Let ( Ω , ℱ , ℚ ) be a risk-neutral probability space supporting two correlated Brownian motions ( W t F , W t v ) with correlation ρ ∈ ( − 1 , 1 ) . Under the risk-neutral measure ℚ , the dynamics of the copper futures price F t and its instantaneous variance v t are given by the classical Heston model:

with

Here, κ > 0 denotes the speed of mean reversion of the variance process, θ > 0 is the long-term variance level, σ > 0 represents the volatility of variance, v 0 > 0 is the initial variance, and ρ captures the leverage effect between futures returns and volatility innovations.

Unlike spot-price models, the futures price process (1) contains no drift term under ℚ , reflecting the martingale property of futures prices in arbitrage-free markets [23].

To ensure the strict positivity of the variance process v t , the parameters are commonly assumed to satisfy the Feller condition:

Although this condition is sufficient but not necessary, it is often imposed in practice to guarantee numerical stability and to avoid variance hitting zero.

Let U ( t , F , v ) denote the price at time t of a European-style contingent claim written on a copper futures contract with maturity T and payoff Φ ( F T ) . Under standard regularity assumptions, U satisfies the Heston partial differential equation:

with terminal condition

Equation (4) provides the foundation for both analytical approaches based on characteristic functions and numerical methods, such as finite differences or Monte Carlo simulation.

Let κ ˜ , θ ˜ , σ ˜ , v ˜ 0 , ρ ˜ denote trapezoidal fuzzy parameters representing uncertainty in the mean-reversion speed, long-term variance, volatility of variance, initial variance, and correlation, respectively. For each α ∈ [ 0 , 1 ] , the α -cuts of these parameters are interval-valued functions

with

and − 1 < ρ ( α ) < 1 . The α -cut decomposition allows us to propagate uncertainty through a family of classical Heston models indexed by α .

Fix α ∈ [ 0 , 1 ] . Under the risk-neutral probability measure ℚ , the α -level copper futures price F t and variance process V t satisfy

with

and initial conditions F 0 = F > 0 , V 0 = v > 0 .

Let us state now on the possedness of the α -Level of the (6) and (7) equations.

Lemma4.1. Well-posedness of the α -Level SDE For each fixed α ∈ [ 0 , 1 ] , the α -level Heston system admits a unique strong solution ( F t , V t ) such that V t ≥ 0 almost surely.

Proof. Since the coefficients satisfy local Lipschitz and linear growth conditions, and κ ( α ) , θ ( α ) , σ ( α ) > 0 , standard results for Cox-Ingersoll-Ross (CIR) processes [26][27] ensure existence, uniqueness, and non-negativity of V t . The futures price process then follows uniquely.

Let Φ : ℝ + → ℝ denote the terminal payoff function, assumed to be Lipschitz continuous with polynomial growth.

Theorem4.1 (Feynman-Kac Representation for the Fuzzy Heston Model). Foreachfixed α ∈ [ 0 , 1 ] , considerthe α -levelHestondynamicsgivenbythe (6) and (7) equations.

Withinitialcondition ( F t , V t ) = ( F , v ) attime t .

Then, the α -levelpriceofaEuropean-stylederivativewithpayoff Φ ( F T ) is

Moreover, U ( t , F , v ; α ) isaclassicalsolutionofthe α -levelpricingPDE

withterminalcondition

Proof. The result follows directly from the classical Feynman-Kac theorem [28][29].

1) For fixed α , the SDE system satisfies local Lipschitz and linear growth conditions (Lemma 4.1), ensuring the existence of a unique strong solution ( F t , V t ) .

2) The function U ( t , F , v ; α ) : = E ℚ [ Φ ( F T ) | F t = F , V t = v ] is well-defined.

3) Applying Itô’s formula to U ( s , F s , V s ; α ) over s ∈ [ t , T ] yields

where ℒ α is the generator of the α -level SDE:

4) Since the martingale term has zero expectation under ℚ , taking expectations and using the terminal condition U ( T , F T , V T ; α ) = Φ ( F T ) gives the PDE (9).

Hence, U ( t , F , v ; α ) is the unique classical solution of the α -level Heston PDE.

This formulation allows us to compute the fuzzy price U ˜ ( t , F , v ) = { U ( t , F , v ; α ) : α ∈ [ 0 , 1 ] } through α -cut decomposition.

Proposition4.2. (Existence and Uniqueness of the α -Level PDE). Foreachfixed α ∈ [ 0 , 1 ] , assumethatthepayofffunction Φ : ℝ + → ℝ isLipschitzcontinuouswithpolynomialgrowth.Thenthe α -levelpricingPDE

withterminalcondition

admitsauniqueclassicalsolution

Proof. Fix α ∈ [ 0 , 1 ] . By Lemma 4.1, the α -level Heston stochastic differential equations admit a unique strong solution ( F t , V t ) with V t ≥ 0 almost surely.

Define

Since Φ is Lipschitz with polynomial growth and the coefficients of the α -level SDE satisfy linear growth conditions, standard moment estimates ensure that U is well defined and finite.

By the classical Feynman-Kac theorem for degenerate parabolic equations [30][31], the function U is a classical solution of the α -level PDE (10).

Uniqueness follows from the parabolic maximum principle applied to (10). In particular, if U 1 and U 2 are two classical solutions with the same terminal condition, their difference satisfies a homogeneous parabolic equation, implying U 1 = U 2 on [ 0 , T ] × ℝ + 2 .

Corollary4.3.Well-definedFuzzyPrice: Thefuzzypricingfunction

iswelldefinedthroughits α -cuts { U ( t , F , v ; α ) } α ∈ [ 0 , 1 ] , anduniquelydeterminedateachconfidencelevel α ∈ [ 0 , 1 ] .

Proof. By Proposition 4.2, for each fixed α ∈ [ 0 , 1 ] the α -level pricing PDE admits a unique classical solution U ( t , F , v ; α ) .

Moreover, the α -cut representation theorem for fuzzy numbers [25][32] ensures that a fuzzy-valued function is uniquely characterized by a nested family of closed intervals indexed by α ∈ [ 0 , 1 ] . Since the family

is well defined and consistent for all α , it uniquely defines the fuzzy pricing function U ˜ ( t , F , v ) .

Corollary4.4.JustificationoftheFuzzyDeepGalerkinMethodSincethe α -levelpricingPDEadmitsauniqueclassicalsolutionforeach α ∈ [ 0 , 1 ] , neural-network-basedsolverssuchastheFuzzyDeepGalerkinMethod (FDGM) converge—whentrainedconsistentlyateach α —tothecorrect α -levelsolution.Consequently, FDGMprovidesaconsistentnumericalapproximationofthefuzzypricingfunction U ˜ ( t , F , v ) .

Proof. For each fixed α , the FDGM approximates the solution of a well-posed, degenerate parabolic PDE with a unique classical solution. Convergence of deep Galerkin-type methods to the true PDE solution follows from standard results on neural-network approximations of parabolic PDEs [18][33].

Since the approximation is performed independently at each confidence level α , and the α -cut solutions uniquely define the fuzzy price by Corollary 4.3, FDGM converges to the correct fuzzy pricing solution.

This Figure 1 provides an intuitive visualization of how fuzzy parameters propagate into price uncertainty, complementing the theoretical well-posedness and numerical convergence results established earlier.

Figure 1. Fuzzy price bands for copper futures. The dark blue line represents the observed futures price, the green line the upper fuzzy bound, and the red line the lower fuzzy bound.

For each fixed α ∈ [ 0 , 1 ] , the α -level pricing function

is approximated by a neural network

where θ denotes the collection of network parameters.

The network input is

and the output is a scalar representing the approximate futures or option price.

Architecture.A fully connected feedforward neural network is employed, consisting of:

One input layer with four neurons, L hidden layers, each with N neurons, One output layer with a single neuron.

The activation function in the hidden layers is chosen as tanh, while the output layer uses a linear activation. Typical values used in the numerical experiments are L = 2   -   5 and N = 40   -   80 .

The FDGM loss function penalizes deviations from the governing α -level PDE as well as violations of the terminal and boundary conditions.

5.2.1. PDE Residual Loss

Let ℒ α denote the differential operator associated with the α -level Heston PDE. The PDE residual loss is defined as

where the expectation is approximated by Monte Carlo sampling over the space-time domain.

5.2.2. Terminal Condition Loss

The terminal payoff condition is enforced through

5.2.3. Boundary Condition Loss

Boundary conditions at extreme values of F and v are weakly enforced via

5.2.4. Total Loss

The total FDGM loss is

where λ TC and λ BC are penalty parameters.

The parameters θ are calibrated by minimizing ℒ ( θ ) using stochastic gradient-based methods.

SamplingStrategy. At each iteration,

Interior collocation points are sampled uniformly in ( t , F , v ) , Terminal points are sampled at t = T , Boundary points are sampled at extreme values of F and v .

Optimizer. The Adam optimizer is employed, with learning-rate decay to improve convergence.

The above procedure is repeated for a discrete set of α -levels,

At each α -level, an independent neural network is trained. The resulting family of solutions provides the lower and upper bounds of the fuzzy price.

Remark 5.1. The independence of the α -level problems allows for straightforward parallel implementation and significantly reduces computational cost.

Let us provide some theorems which will ensure the convergence of the neural network to the fuzzy solution (fuzzy parameters).

Theorem5.2 Well-posedness of the α -level PDE. For each fixed α ∈ [ 0 , 1 ] , the α -level Heston-type PDE admits a unique viscosity solution with polynomial growth.

Proof. For fixed α , the fuzzy parameters reduce to deterministic coefficients. The result follows from standard viscosity solution theory for degenerate parabolic equations; see, e.g., Pham [31] or Fleming and Soner [34].

Theorem5.3 Consistency of the Fuzzy Deep Galerkin Method. Let u ( ⋅ , ⋅ ; α ) be the unique viscosity solution of the α -level PDE. Then the FDGM loss functional satisfies

Proof. If the loss vanishes, the PDE residual and all constraints are satisfied almost everywhere, implying that u coincides with the unique viscosity solution. Conversely, inserting the exact solution yields zero loss.

Theorem5.4 Convergence at Fixed α . Let { u α ( n ) } n ≥ 1 be a sequence of neural networks minimizing the DGM loss. Then

Proof. Density of neural networks in C ( K ) and stability of viscosity solutions with respect to residual perturbations imply convergence; see Sirignano and Spiliopoulos [18].

Theorem5.5 Convergence to the Fuzzy Solution. The family { u α ( n ) } α ∈ [ 0 , 1 ] converges to the fuzzy solution u ˜ in the sense of α -cuts.

Proof. Convergence holds pointwise for each α -level. By the representation theorem for fuzzy numbers, the limit family defines a unique fuzzy-valued solution.

Remark 5.6. The FDGM framework is thus consistent, stable, and convergent for fuzzy pricing problems. While a complete optimization-level convergence proof remains challenging, theoretical well-posedness and strong numerical evidence support the reliability of the method.

We consider derivatives written on futures contracts rather than the futures prices themselves. Under the risk-neutral measure, the futures price process ( F t ) t ≥ 0 is a martingale, and therefore its expectation does not depend on model parameters.

However, option prices written on futures depend on the full distribution of F T , which is governed by the stochastic variance dynamics. In this work, we assume that the futures price follows a Heston-type stochastic volatility model, and we calibrate the model parameters using observed option prices (or fuzzy prices obtained via uncertainty propagation). This explains why the calibrated quantities depend on the Heston parameters despite the martingale property of futures.

To ensure the well-posedness of the stochastic volatility model, we impose feasibility constraints on all α -cuts of the fuzzy parameters. For each α ∈ [ 0 , 1 ] , we require

In addition, we enforce the Feller condition 2 κ ( α ) θ ( α ) ≥ ( σ ( α ) ) 2 , which guarantees strict positivity of the variance process.

These constraints are enforced consistently across all α -levels during calibration.

The FDGM is trained on a truncated computational domain defined by F ∈ [ F min , F max ] ,     v ∈ [ 0 , v max ] ,     t ∈ [ 0 , T ] .

The terminal condition is given by the payoff function U ( T , F , v ) = Φ ( F ) , where Φ denotes the payoff of the option.

Boundary conditions are imposed as follows:

Dirichlet-type conditions at F min and F max Degenerate condition at v = 0 Neumann-type condition at v = v max

Training points are sampled as:

Interior points uniformly in ( t , F , v ) Boundary points on domain boundaries Terminal points at t = T

We consider a dataset of copper futures contracts traded on an organized commodity exchange, covering multiple maturities

For each maturity T i , the corresponding futures settlement price

is observed.

In practice, commodity futures prices are subject to market frictions, bid-ask spreads, and liquidity effects, which introduce uncertainty around quoted prices [1][2]. To account for this uncertainty, observed futures prices are represented as trapezoidal fuzzy numbers

whose α -cuts

reflect confidence intervals around the quoted settlement prices. This representation naturally captures market ambiguity while remaining consistent with empirical pricing data.

The fuzzy parameters to be calibrated are

where κ ˜ denotes the fuzzy mean-reversion speed, θ ˜ the fuzzy long-term variance, σ ˜ the fuzzy volatility of variance, v ˜ 0 the fuzzy initial variance, and ρ ˜ the fuzzy correlation parameter.

For each fixed α ∈ [ 0 , 1 ] , these fuzzy parameters reduce to deterministic α -level values

which are employed in the corresponding α -level pricing PDE. This α -cut-based calibration strategy ensures consistency with the extension principle of fuzzy set theory [8][21].

Figure 2 compares observed copper futures prices with the FDGM-calibrated fuzzy Heston mid-price at α = 1 . The close agreement across maturities demonstrates the ability of the proposed fuzzy framework to accurately capture the term structure of copper futures prices. Compared to classical pointwise calibration, the fuzzy approach provides additional information on parameter robustness and uncertainty.

Figure 2. Comparison between observed copper futures prices and FDGM-calibrated fuzzy Heston prices. Discrete markers represent market prices, while the solid curve corresponds to the fuzzy mid-price at α = 1 .

At each α -level, the calibration problem is formulated as a least-squares minimization problem. Let

denote the futures price generated by the fuzzy Heston model using FDGM. The α -level objective function is defined as

This formulation is consistent with standard calibration practices for stochastic volatility models [37] while naturally extending them to the fuzzy setting.

The calibration is performed sequentially over the discrete set of α -levels

For each α :

1) An initial guess for Θ ( α ) is selected based on historical estimates and values reported in the literature.

2) The FDGM neural network is trained to solve the α -level pricing PDE.

3) Model parameters are updated by minimizing the objective function J ( Θ ( α ) ) .

Gradient-based optimization using the Adam optimizer is employed due to its robustness in high-dimensional non-convex problems [38]. Parameter constraints

are enforced through suitable parameter transformations.

As described above, our approach relies on the following procedure:

Algorithm 1 summarizes the proposed Fuzzy Deep Galerkin Method (FDGM) for solving α -level partial differential equations arising from fuzzy extensions of stochastic volatility models, providing a systematic procedure for network initialization, α -cut propagation, loss evaluation, and optimization.

Algorithm 1.Fuzzy Deep Galerkin Method (FDGM).

The calibration is performed through a joint optimization procedure involving both the neural network parameters and the Heston model parameters.

At each iteration, the following steps are performed:

1) The neural network parameters are updated by minimizing the FDGM loss function defined in Equation (14).

2) The Heston parameters are updated by minimizing the calibration objective defined in Equation (15), based on the discrepancy between model prices and observed prices.

This alternating optimization procedure ensures consistency between the PDE solution provided by the FDGM and the calibration objective.

The calibration yields a family of parameter estimates

which collectively define the fuzzy Heston model. The resulting fuzzy parameter bands capture the uncertainty inherent in the copper futures market and provide a confidence-dependent description of stochastic volatility dynamics.

To assess calibration quality, we compute the Fuzzy Mean Squared Error (FMSE) at each α -level:

This metric allows for a direct comparison of model accuracy across confidence levels and highlights the robustness of the fuzzy calibration.

We observe that the long-term variance θ and volatility of variance σ increase slightly with maturity, indicating a term structure effect in copper market volatility. The width of the fuzzy bands remains relatively stable, suggesting consistent model uncertainty across maturities. This visualization validates that the FDGM calibration produces plausible and interpretable fuzzy parameter estimates.

The calibration results indicate that the fuzzy Heston model achieves an accurate fit to copper futures prices across maturities and α -levels. As expected, the FMSE decreases as α increases, reflecting tighter uncertainty bands near the core of the fuzzy set.

Overall, the proposed FDGM-based fuzzy calibration framework combines numerical efficiency with a rigorous treatment of uncertainty. It extends classical stochastic volatility calibration by providing parameter confidence bands and enhanced robustness, making it particularly suitable for commodity markets characterized by structural uncertainty and regime variability [1][2].

The FDGM calibration is performed for

For each α -level, the calibrated model produces a futures price

which is compared to the corresponding fuzzy market observation

Table 1 reports the Fuzzy Mean Squared Error (FMSE) across α -levels.

As expected, the FMSE decreases monotonically with α , indicating that model precision improves as the confidence level increases. This behavior is consistent with fuzzy set theory, where α = 1 corresponds to the most plausible parameter configuration, while lower α -levels capture extreme but less likely scenarios (Figure 3).

Table 1. Fuzzy Mean Squared Error (FMSE) across α -levels for copper futures.

Figure 3. Fuzzy Mean Squared Error (FMSE) across α -levels for copper futures. The monotonic decrease of FMSE as α increases reflects improved pricing accuracy as uncertainty bands contract toward the core of the fuzzy set, confirming the robustness of the FDGM calibration.

To assess the added value of fuzziness and the FDGM framework, the proposed model is compared with the following benchmarks:

The Black-Scholes model with constant volatility, The classical Heston model calibrated via Monte Carlo simulation, The classical Heston model calibrated via FFT-based methods.

Table 2 reports the Mean Squared Error (MSE) at α = 1 .

The fuzzy Heston model calibrated via FDGM achieves the lowest error, highlighting the benefits of incorporating parameter uncertainty and solving the pricing equations directly through a mesh-free neural framework. The results suggest that classical pointwise calibration underestimates structural uncertainty in commodity markets.

Table 2. Model comparison at α = 1 .

Figure 4 illustrates the fuzzy copper futures price as a function of maturity T and confidence level α . For each fixed maturity, the futures price uncertainty contracts monotonically as α increases, converging to the core price at α = 1 . Along the maturity dimension, the surface exhibits a clear term-structure effect, with uncertainty becoming more pronounced at longer maturities. This three-dimensional representation provides an intuitive visualization of how trapezoidal fuzzy parameter uncertainty propagates through the Heston dynamics into maturity-dependent price uncertainty.

Figure 4. Three-dimensional fuzzy copper futures price surface as a function of maturity T and confidence level α .

Table 3 reports the trapezoidal fuzzy Heston parameters calibrated to copper futures prices for different maturities. As maturity increases, the speed of mean reversion slightly decreases while the volatility-of-volatility and long-run variance exhibit wider fuzzy spreads, reflecting increased uncertainty at longer horizons. The fuzzy correlation parameter remains strongly negative across maturities, consistent with the leverage effect observed in commodity markets.

Table 3. Trapezoidal fuzzy Heston parameters calibrated to copper futures prices. (a) Maturity and fuzzy parameters κ ˜ , θ ˜ , and σ ˜ ; (b) Maturity and fuzzy parameters v ˜ 0 and ρ ˜ .

Sensitivity analysis is conducted by perturbing each fuzzy parameter within its α -cut while holding the remaining parameters fixed. The results indicate that the model is:

Highly sensitive to the long-term variance θ and volatility of variance σ , Moderately sensitive to the correlation parameter ρ , Relatively robust to small perturbations in κ and v 0 .

These findings are consistent with empirical evidence in commodity markets, where volatility dynamics dominate price formation.

To assess numerical robustness, the calibration procedure is repeated using:

Different initial parameter guesses, Subsampled futures datasets, Synthetic noise added to market prices.

Across all experiments, the FDGM converges to stable parameter estimates and maintains low FMSE values, demonstrating strong robustness and resilience to data perturbations.

Figure 5 reports the evolution of the DGM loss function for networks with 2, 3, 4, and 5 hidden layers. Increasing network depth accelerates convergence and yields lower terminal loss values, reflecting enhanced approximation capacity. Deeper networks achieve lower residuals, although marginal improvements diminish beyond a certain depth, highlighting a trade-off between accuracy and computational cost.

Figure 5. Loss function evolution for the Deep Galerkin Method applied to the fuzzy Heston PDE.

Despite solving multiple PDEs across α -levels, the FDGM framework remains computationally efficient due to:

Parallelizable α -level training,Mesh-free sampling, Avoidance of high-dimensional grids.

This scalability makes the proposed approach suitable for large-scale calibration problems in commodity markets.