Finance-Informed Neural Networks (FINNs)

• FINNs are neural-network pricing frameworks that integrate financial theory, embedding no-arbitrage and PDE constraints directly into model training.
• They combine deterministic PDE methods with data-driven approaches to price derivatives under Black-Scholes, Heston, and HJM models.
• Empirical studies demonstrate that FINNs achieve high accuracy and significant speedups over Monte Carlo methods in high-dimensional settings.

Finance-Informed Neural Networks (FINNs) are neural-network pricing frameworks that embed no-arbitrage structure directly into model training rather than treating derivative valuation as a purely supervised regression task. In the cited literature, the term denotes hybrid methodologies that combine the theoretical rigor and interpretability of PDE-based pricing with the adaptability of machine learning, either by minimizing violations of a pricing PDE on a high-dimensional state space or by minimizing discrete self-financing hedging residuals that enforce the continuous-time pricing equation in the limit (Mott, 12 Mar 2026, Aboussalah et al., 2024). As presented in these works, FINNs are used for option pricing under Black-Scholes and Heston dynamics and for path-dependent interest-rate derivatives under a discretized Heath-Jarrow-Morton (HJM) forward-curve model.

1. Conceptual scope

The two cited formulations share a common organizing principle: the pricing function is parameterized by a feed-forward neural network, while financial structure is imposed through no-arbitrage dynamics, PDE constraints, hedging identities, and terminal or boundary conditions. One formulation is explicitly introduced as a hybrid approach integrating “principle-driven methods” and “data-driven approaches,” with the aim of preserving theoretical consistency while improving adaptability across market conditions (Aboussalah et al., 2024). The other casts an infinite-dimensional HJM pricing problem as a deterministic PDE and solves it directly with a neural network that minimizes violations of the differential equation and boundary condition, yielding a “Monte Carlo-free approach” to pricing path-dependent interest-rate derivatives (Mott, 12 Mar 2026).

Variant	Governing structure	Representative instruments
Black-Scholes/Heston FINN	Self-financing hedging residual and PDE consistency	Calls; delta-gamma hedging extension
HJM FINN	Backward Feynman-Kac PDE on the full forward curve	Caplets, caps, swaptions, callable bonds

A plausible implication is that “FINN” is best understood not as a single fixed architecture, but as a family of finance-constrained neural solvers whose defining feature is the direct incorporation of pricing theory into the loss functional.

2. Governing equations and no-arbitrage structure

In the HJM setting, the infinite forward curve $f(t,\tau)$ is discretized into $K$ tenors $\{\tau_k\}_{k=1}^K$, with state vector

$$f(t)=\bigl(f_1(t),\dots,f_K(t)\bigr)^\top,\qquad f_k(t)=f(t,\tau_k).$$

Under the Musiela parameterization and risk-neutral measure,

$$df(t)=\mu\bigl(t,f(t)\bigr)\,dt+\sum_{n=1}^N\sigma_n\bigl(t,f(t)\bigr)\,dW_n(t),$$

where

$$\mu_k(t,f)=\frac{\partial}{\partial\tau}f(t,\tau_k)+\sigma(t,\tau_k,f)\int_0^{\tau_k}\sigma(t,s,f)\,ds,$$

$\sigma_n(t,f)\in\mathbb R^K$ is the $n$th volatility factor, and the short rate is $r(t)=f_1(t)$. By the multidimensional Feynman-Kac theorem, the time-$t$ price

$K$0

solves

$K$1

with terminal condition $K$2 (Mott, 12 Mar 2026).

A notable feature of this formulation is the treatment of path dependence. The original problem is path-dependent because discounting requires $K$3 and because the payoff depends on the full forward-curve realization at $K$4. In the PDE, these channels are retained through the $K$5 term and the terminal boundary condition. The summary explicitly states that no additional auxiliary state variables are required; the dependence on the entire past path appears through solving $K$6 on the full $K$7-dimensional curve space (Mott, 12 Mar 2026). This directly addresses a common misconception that path dependence must always be handled by augmenting the state with explicit path integrals.

In the equity-option formulation, FINN is tied to the standard pricing PDEs. For Black-Scholes,

$K$8

while under Heston dynamics,

$K$9

with $\{\tau_k\}_{k=1}^K$0 (Aboussalah et al., 2024). In that framework, the PDE is enforced through a dynamic-hedging identity: a self-financing, zero-cost, riskless portfolio is constructed so that the residual vanishes under no arbitrage.

3. Neural parameterizations and input design

The HJM formulation uses a network with inputs $\{\tau_k\}_{k=1}^K$1 consisting of $\{\tau_k\}_{k=1}^K$2 discretized forward rates $\{\tau_k\}_{k=1}^K$3, six Svensson parameters $\{\tau_k\}_{k=1}^K$4, each z-scored, and three contract features $\{\tau_k\}_{k=1}^K$5 normalized by $\{\tau_k\}_{k=1}^K$6. The output $\{\tau_k\}_{k=1}^K$7 is the scalar price $\{\tau_k\}_{k=1}^K$8. The hidden architecture comprises three fully-connected layers of width 500 with SiLU activations, and the output activation is softplus to enforce $\{\tau_k\}_{k=1}^K$9. Feature-wise normalization is built into the first layer, while forward rates remain unnormalized to preserve their economic scale (Mott, 12 Mar 2026).

The Black-Scholes/Heston formulation adopts a smaller feed-forward network. At each time $$f(t)=\bigl(f_1(t),\dots,f_K(t)\bigr)^\top,\qquad f_k(t)=f(t,\tau_k).$$0, the input is

$$f(t)=\bigl(f_1(t),\dots,f_K(t)\bigr)^\top,\qquad f_k(t)=f(t,\tau_k).$$1

and may include additional state variables such as instantaneous variance $$f(t)=\bigl(f_1(t),\dots,f_K(t)\bigr)^\top,\qquad f_k(t)=f(t,\tau_k).$$2. The experiments reported use $$f(t)=\bigl(f_1(t),\dots,f_K(t)\bigr)^\top,\qquad f_k(t)=f(t,\tau_k).$$3 hidden layers of width $$f(t)=\bigl(f_1(t),\dots,f_K(t)\bigr)^\top,\qquad f_k(t)=f(t,\tau_k).$$4, with fully connected layers, $$f(t)=\bigl(f_1(t),\dots,f_K(t)\bigr)^\top,\qquad f_k(t)=f(t,\tau_k).$$5 hidden activations, and a softplus output so that the learned quantity $$f(t)=\bigl(f_1(t),\dots,f_K(t)\bigr)^\top,\qquad f_k(t)=f(t,\tau_k).$$6 remains positive. In the GBM case, the network parameterizes the pricing map by

$$f(t)=\bigl(f_1(t),\dots,f_K(t)\bigr)^\top,\qquad f_k(t)=f(t,\tau_k).$$7

so that the model directly approximates $$f(t)=\bigl(f_1(t),\dots,f_K(t)\bigr)^\top,\qquad f_k(t)=f(t,\tau_k).$$8 (Aboussalah et al., 2024).

These architectural choices indicate that the “finance-informed” designation does not depend on a single depth, width, or activation family. The common feature is the coupling of a positive price network with economically structured inputs and theory-constrained losses.

4. Loss construction, automatic differentiation, and training

In the HJM FINN, the core training signal is the PDE residual

$$f(t)=\bigl(f_1(t),\dots,f_K(t)\bigr)^\top,\qquad f_k(t)=f(t,\tau_k).$$9

The minibatch loss combines three terms: the squared PDE residual, the terminal boundary penalty for positive-strike caplets, and a zero-strike anchoring penalty based on the analytical closed form. The payoff at $$df(t)=\mu\bigl(t,f(t)\bigr)\,dt+\sum_{n=1}^N\sigma_n\bigl(t,f(t)\bigr)\,dW_n(t),$$0 is

$$df(t)=\mu\bigl(t,f(t)\bigr)\,dt+\sum_{n=1}^N\sigma_n\bigl(t,f(t)\bigr)\,dW_n(t),$$1

while the closed-form zero-strike caplet is

$$df(t)=\mu\bigl(t,f(t)\bigr)\,dt+\sum_{n=1}^N\sigma_n\bigl(t,f(t)\bigr)\,dW_n(t),$$2

Automatic differentiation provides exact, machine-precision values for $$df(t)=\mu\bigl(t,f(t)\bigr)\,dt+\sum_{n=1}^N\sigma_n\bigl(t,f(t)\bigr)\,dW_n(t),$$3, $$df(t)=\mu\bigl(t,f(t)\bigr)\,dt+\sum_{n=1}^N\sigma_n\bigl(t,f(t)\bigr)\,dW_n(t),$$4, and directional second derivatives; the identity

$$df(t)=\mu\bigl(t,f(t)\bigr)\,dt+\sum_{n=1}^N\sigma_n\bigl(t,f(t)\bigr)\,dW_n(t),$$5

avoids constructing the full Hessian and is stated to be computable in $$df(t)=\mu\bigl(t,f(t)\bigr)\,dt+\sum_{n=1}^N\sigma_n\bigl(t,f(t)\bigr)\,dW_n(t),$$6 time. The implementation discussion explicitly names JAX, TensorFlow, and PyTorch as platforms on which these derivatives can be obtained via AD rather than finite differences (Mott, 12 Mar 2026).

Training in that setting uses Adam with weight decay $$df(t)=\mu\bigl(t,f(t)\bigr)\,dt+\sum_{n=1}^N\sigma_n\bigl(t,f(t)\bigr)\,dW_n(t),$$7 and a three-phase curriculum: Phase 1 with 15,000 epochs at learning rate $$df(t)=\mu\bigl(t,f(t)\bigr)\,dt+\sum_{n=1}^N\sigma_n\bigl(t,f(t)\bigr)\,dW_n(t),$$8, batch size 100, and 10 batches per epoch; Phase 2 with 5,000 epochs at learning rate $$df(t)=\mu\bigl(t,f(t)\bigr)\,dt+\sum_{n=1}^N\sigma_n\bigl(t,f(t)\bigr)\,dW_n(t),$$9, batch size 100, and 10 batches per epoch; and Phase 3 with 2,500 epochs at learning rate $$\mu_k(t,f)=\frac{\partial}{\partial\tau}f(t,\tau_k)+\sigma(t,\tau_k,f)\int_0^{\tau_k}\sigma(t,s,f)\,ds,$$0, batch size 500, and 2 batches per epoch. The sampling strategy draws $$\mu_k(t,f)=\frac{\partial}{\partial\tau}f(t,\tau_k)+\sigma(t,\tau_k,f)\int_0^{\tau_k}\sigma(t,s,f)\,ds,$$1 uniformly in typical caplet ranges, samples a historical forward curve and Svensson parameters, and selects strike $$\mu_k(t,f)=\frac{\partial}{\partial\tau}f(t,\tau_k)+\sigma(t,\tau_k,f)\int_0^{\tau_k}\sigma(t,s,f)\,ds,$$2 from a Chebyshev grid on $$\mu_k(t,f)=\frac{\partial}{\partial\tau}f(t,\tau_k)+\sigma(t,\tau_k,f)\int_0^{\tau_k}\sigma(t,s,f)\,ds,$$3. Precomputations include a trapezoidal-rule integration matrix for bond-price and drift integrals, Chebyshev coefficients of PCA-derived $$\mu_k(t,f)=\frac{\partial}{\partial\tau}f(t,\tau_k)+\sigma(t,\tau_k,f)\int_0^{\tau_k}\sigma(t,s,f)\,ds,$$4, and index lists of admissible $$\mu_k(t,f)=\frac{\partial}{\partial\tau}f(t,\tau_k)+\sigma(t,\tau_k,f)\int_0^{\tau_k}\sigma(t,s,f)\,ds,$$5 pairs (Mott, 12 Mar 2026).

In the Black-Scholes/Heston FINN, the loss originates from a discrete self-financing hedging relation. With $$\mu_k(t,f)=\frac{\partial}{\partial\tau}f(t,\tau_k)+\sigma(t,\tau_k,f)\int_0^{\tau_k}\sigma(t,s,f)\,ds,$$6 obtained by auto-differentiation and $$\mu_k(t,f)=\frac{\partial}{\partial\tau}f(t,\tau_k)+\sigma(t,\tau_k,f)\int_0^{\tau_k}\sigma(t,s,f)\,ds,$$7, the residual is

$$\mu_k(t,f)=\frac{\partial}{\partial\tau}f(t,\tau_k)+\sigma(t,\tau_k,f)\int_0^{\tau_k}\sigma(t,s,f)\,ds,$$8

Over a minibatch $$\mu_k(t,f)=\frac{\partial}{\partial\tau}f(t,\tau_k)+\sigma(t,\tau_k,f)\int_0^{\tau_k}\sigma(t,s,f)\,ds,$$9, the overall loss is the average of this residual, with an optional delta-gamma term

$\sigma_n(t,f)\in\mathbb R^K$0

The summary stresses that no supervised “market-price” labels are required; training proceeds solely by minimizing hedging/PDE residuals. Data are generated by simulating GBM or Heston paths via Euler-Maruyama with $\sigma_n(t,f)\in\mathbb R^K$1 such as $\sigma_n(t,f)\in\mathbb R^K$2, drawing random $\sigma_n(t,f)\in\mathbb R^K$3 and $\sigma_n(t,f)\in\mathbb R^K$4, and forming tuples of $\sigma_n(t,f)\in\mathbb R^K$5. Reported optimization settings are Adam with learning rate $\sigma_n(t,f)\in\mathbb R^K$6, mini-batch sizes 64–256 with 128 used in the experiments, 100–500 epochs, early stopping on a held-out validation set, and cross-validation for the gamma-loss weight $\sigma_n(t,f)\in\mathbb R^K$7 (Aboussalah et al., 2024).

5. Reported empirical behavior

For the HJM caplet problem, the test set contains 1,000 random caplets per discretization $\sigma_n(t,f)\in\mathbb R^K$8, and the Monte Carlo benchmark uses 10,000 paths under the same local-volatility HJM. The reported mean absolute errors in cents per dollar are

$\sigma_n(t,f)\in\mathbb R^K$9

and the summary states that all errors lie between roughly $n$0 ¢ and $n$1 ¢ per \$n$22\times 10^{{-6}$n$36\times} 10<sup>{-6}$n$4K$n$5K=10$n$6K=150$n$7300{,}000\times$n$84.5$n$9\times$ (Mott, 12 Mar 2026).

The same HJM formulation also reports that theta, $r(t)=f_1(t)$0, and curve deltas, $r(t)=f_1(t)$1, are direct outputs of PDE residual evaluation and therefore available at zero marginal cost. Higher-order Greeks such as gamma and vega require additional AD calls but are still described as orders of magnitude faster than re-simulating Monte Carlo (Mott, 12 Mar 2026).

For the Black-Scholes/Heston formulation, the evaluation criteria are price error and hedge-ratio error, each measured by MAD and MSE against analytic or Monte Carlo benchmarks. For 2-month calls with $r(t)=f_1(t)$2 over $r(t)=f_1(t)$3 and $r(t)=f_1(t)$4, averaged over 10 runs, the reported values are price MAD $r(t)=f_1(t)$5 and delta MAD $r(t)=f_1(t)$6, with accuracy said to remain well under 1–2% relative error in price and $r(t)=f_1(t)$7 as $r(t)=f_1(t)$8 or $r(t)=f_1(t)$9 increase. Under Heston with vol-of-vol $t$0 for 2-month calls, price MAD is reported as $t$1–$t$2 and delta MAD as $t$3–$t$4. In the delta-gamma extension using an ATM option as hedging instrument for a 2-month call, price MAD falls from $t$5 to $t$6 at $t$7, and gamma MAD is $t$8. The forward and gradient passes of the $t$9 network require approximately $K$00 seconds per evaluation on a GPU (Aboussalah et al., 2024).

Taken together, these results suggest two different computational advantages. In the interest-rate setting, the dominant claim is extreme post-training speed relative to Monte Carlo in a high-dimensional state space. In the option setting, the dominant claim is that no-arbitrage self-supervision yields competitive prices and hedge ratios without label-based training.

6. Extensions, limitations, and interpretive issues

The HJM FINN is described as generalizing naturally beyond caplets. Caps are treated as portfolios of caplets, so one may train once and evaluate with multiple $K$01, or modify the terminal penalty to the sum of individual caplet payoffs. Swaptions replace the terminal payoff $K$02 by the discounted payoff of the swap rate at exercise while retaining the same PDE and network. Callable bonds and other path-dependent features are introduced by changing only the boundary condition $K$03 and, if early exercise is involved, adding an obstacle term, while leaving the drift and volatility terms untouched (Mott, 12 Mar 2026).

The option-pricing FINN is likewise presented as extensible. It can be retrained on any risk-neutral SDE, including jump-diffusion, local volatility, and rough volatility; American and path-dependent or exotic options can be handled by incorporating early-exercise via dynamic or free-boundary terms in the loss; and multi-asset basket options are treated by increasing the input dimension while preserving the same self-supervision loss that enforces the multidimensional PDE (Aboussalah et al., 2024).

Several interpretive clarifications follow directly from the cited material. First, FINN is not synonymous with supervised price fitting: in the Black-Scholes/Heston formulation, no supervised market-price labels are required. Second, “Monte Carlo-free” is specific to the HJM pricing approach after reformulating the problem as a deterministic PDE; it does not imply that all FINN variants avoid simulation during training, since the option-pricing formulation explicitly generates asset paths by Euler-Maruyama. Third, path dependence does not necessarily require explicit auxiliary path states in the HJM formulation, because the full discretized curve space serves as the state domain (Mott, 12 Mar 2026, Aboussalah et al., 2024).

The limitations stated in the HJM summary are also concrete. There is an up-front cost of network training of approximately one hour on an 8 GB GPU. Accuracy is non-monotonic in the forward-curve discretization, which is said to suggest interaction with network capacity, so fine-tuning may be needed per $K$04. Early-exercise features of American style require additional obstacle or enforcement terms (Mott, 12 Mar 2026). In the option-pricing formulation, training stability is addressed by clipping $K$05 and $K$06 within theoretical bounds, specifically $K$07 and $K$08 (Aboussalah et al., 2024). A plausible implication is that the practical deployment of FINNs depends not only on asymptotic no-arbitrage consistency, but also on regularization, sampling design, and the numerical conditioning of derivative computations.