---
title: Finance-Informed Neural Networks (FINNs)
url: https://www.emergentmind.com/topics/finance-informed-neural-networks-finns
type: topic
---

# Finance-Informed Neural Networks (FINNs)

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 [2603.12375, 2412.12213]. 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 [2412.12213]. 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 [2603.12375].

| 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
$$
V(t,f)=\mathbb E\Bigl[e^{-\int_t^T r(s)ds}\,h\bigl(f(T)\bigr)\,\big|\,\mathcal F_t\Bigr]
$$
solves
$$
\frac{\partial V}{\partial t}
+\mu(t,f)^\top\nabla_f V
+\tfrac12\sum_{n=1}^N \sigma_n(t,f)^\top D_f^2V\,\sigma_n(t,f)
-r(t)V
=0,
$$
with terminal condition $V(T,f)=h(f;\Xi)$ [2603.12375].

A notable feature of this formulation is the treatment of path dependence. The original problem is path-dependent because discounting requires $\int_t^T r(s)\,ds$ and because the payoff depends on the full forward-curve realization at $T$. In the PDE, these channels are retained through the $-rV$ 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 $f\mapsto V$ on the full $K$-dimensional curve space [2603.12375]. 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,
$$
\partial_t u + rS\,\partial_S u + \tfrac12\sigma^2 S^2\,\partial_{SS}u - ru = 0,\qquad
u(T,S)=\max\{S-K,0\},
$$
while under Heston dynamics,
$$
\partial_t u + rS\,\partial_S u
+ \kappa(\theta-\nu)\,\partial_\nu u
+ \tfrac12\nu S^2\,\partial_{SS}u
+ \rho\xi\nu S\,\partial_{S\nu}u
+ \tfrac12\xi^2\nu\,\partial_{\nu\nu}u
- ru = 0,
$$
with $u(T,S,\nu)=\text{payoff}$ [2412.12213]. 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 ${\cal X}\in\mathbb R^{K+9}$ consisting of $K$ discretized forward rates $f$, six Svensson parameters $S=(\beta_0,\dots,\beta_3,\tau_1,\tau_2)$, each z-scored, and three contract features $\Xi=(\tau,\delta,L_E)$ normalized by $\tau_{\max}$. The output ${\cal Y}\in\mathbb R$ is the scalar price $V_\Theta(f,\Xi)$. The hidden architecture comprises three fully-connected layers of width 500 with SiLU activations, and the output activation is softplus to enforce $V\ge 0$. Feature-wise normalization is built into the first layer, while forward rates remain unnormalized to preserve their economic scale [2603.12375].

The Black-Scholes/Heston formulation adopts a smaller feed-forward network. At each time $t$, the input is
$$
I_t=\{ \text{normalized spot } S_t/K\cdot e^{-r\tau},\ \tau=T-t \},
$$
and may include additional state variables such as instantaneous variance $\nu_t$. The experiments reported use $L=2$ hidden layers of width $M=50$, with fully connected layers, $\tanh$ hidden activations, and a softplus output so that the learned quantity $g^\theta(I_t)$ remains positive. In the GBM case, the network parameterizes the pricing map by
$$
u(t,S_t)\equiv g^\theta(I_t),
$$
so that the model directly approximates $(t,S)\mapsto \text{option price}$ [2412.12213].

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
$$
R[\Theta]
= -\frac{\partial V_\Theta}{\partial \tau}
+\mu(\tau,f)^\top\nabla_f V_\Theta
+\tfrac12\sum_{n=1}^N \sigma_n(\tau,f)^\top D_f^2V_\Theta\,\sigma_n(\tau,f)
-r(\tau)V_\Theta.
$$
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 $\tau=0$ is
$$
h(f;\Xi)=\delta\,P(\delta)\max\{L(f)-L_E,0\},
$$
while the closed-form zero-strike caplet is
$$
h_{\mathrm{zero}}(\tau,f)=P(\tau)-P(\tau+\delta).
$$
Automatic differentiation provides exact, machine-precision values for $\nabla_fV_\Theta$, $\partial_\tau V_\Theta$, and directional second derivatives; the identity
$$
\sigma_n^\top D_f^2V\,\sigma_n
= \nabla_f\bigl(\nabla_fV\cdot \sigma_n\bigr)\cdot \sigma_n
$$
avoids constructing the full Hessian and is stated to be computable in $O(K)$ 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 [2603.12375].

Training in that setting uses Adam with weight decay $10^{-5}$ and a three-phase curriculum: Phase 1 with 15,000 epochs at learning rate $10^{-4}$, batch size 100, and 10 batches per epoch; Phase 2 with 5,000 epochs at learning rate $10^{-5}$, batch size 100, and 10 batches per epoch; and Phase 3 with 2,500 epochs at learning rate $10^{-6}$, batch size 500, and 2 batches per epoch. The sampling strategy draws $(\tau,\delta)$ uniformly in typical caplet ranges, samples a historical forward curve and Svensson parameters, and selects strike $L_E$ from a Chebyshev grid on $[0,0.07]$. Precomputations include a trapezoidal-rule integration matrix for bond-price and drift integrals, Chebyshev coefficients of PCA-derived $\sigma_n(\tau)$, and index lists of admissible $(\tau,\delta)$ pairs [2603.12375].

In the Black-Scholes/Heston FINN, the loss originates from a discrete self-financing hedging relation. With $\Delta_t=\partial_S u^\theta(t,S_t)$ obtained by auto-differentiation and $du_t^\theta\approx u^\theta(t+\Delta t)-u_t^\theta$, the residual is
$$
R_t(\theta)
=
\Bigl[
\Delta_t\,dS_t
+r\,(u_t^\theta-\Delta_t S_t)\,dt
-\bigl(u_{t+\Delta t}^\theta-u_t^\theta\bigr)
\Bigr]^2.
$$
Over a minibatch $\mathcal B$, the overall loss is the average of this residual, with an optional delta-gamma term
$$
L_\Gamma(\theta)=
\lambda\bigl[\partial_{SS}u^\theta(t,S_t)-\Gamma_{\rm ref}(t,S_t)\bigr]^2.
$$
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 $\Delta t$ such as $1/250$, drawing random $K$ and $T$, and forming tuples of $(I_t,I_{t+1},dS_t,r,K)$. Reported optimization settings are Adam with learning rate $\sim 10^{-3}$, 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 $\lambda$ [2412.12213].

## 5. Reported empirical behavior

For the HJM caplet problem, the test set contains 1,000 random caplets per discretization $K$, 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
$$
\begin{array}{c|cccccccc}
K & 10 & 25 & 35 & 50 & 75 & 100 & 125 & 150\\\hline
\mathrm{MAE}
&8.73\times 10^{-4}
&6.25\times 10^{-4}
&5.69\times 10^{-4}
&5.17\times 10^{-4}
&9.42\times 10^{-4}
&3.75\times 10^{-4}
&7.98\times 10^{-4}
&4.03\times 10^{-4}
\end{array}
$$
and the summary states that all errors lie between roughly $0.04$ ¢ and $0.10$ ¢ per \$1 notional. Evaluation time is reported as $2\times 10^{-6}$ to $6\times 10^{-6}$ seconds per contract, essentially flat in $K$, while Monte Carlo time grows from approximately 1 second at $K=10$ to approximately 10 seconds at $K=150$. The corresponding speedup ratio ranges from $300{,}000\times$ to $4.5$ million$\times$ [2603.12375].

The same HJM formulation also reports that theta, $-\partial_\tau V_\Theta$, and curve deltas, $\nabla_fV_\Theta$, 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 [2603.12375].

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 $\sigma=0.15$ over $S\in[75,125]$ and $K\in[90,110]$, averaged over 10 runs, the reported values are price MAD $\simeq 0.18$ and delta MAD $\simeq 0.022$, with accuracy said to remain well under 1–2% relative error in price and $\Delta$ as $\sigma$ or $T$ increase. Under Heston with vol-of-vol $\xi\in\{0.125,0.175\}$ for 2-month calls, price MAD is reported as $\simeq 0.26$–$0.38$ and delta MAD as $\simeq 0.03$–$0.04$. In the delta-gamma extension using an ATM option as hedging instrument for a 2-month call, price MAD falls from $\simeq 0.18$ to $\simeq 0.024$ at $\sigma=0.125$, and gamma MAD is $\simeq 0.001$. The forward and gradient passes of the $2\times 50$ network require approximately $10^{-4}$ seconds per evaluation on a GPU [2412.12213].

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 $L_E$, or modify the terminal penalty to the sum of individual caplet payoffs. Swaptions replace the terminal payoff $h(f)$ 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 $h(f;\Xi)$ and, if early exercise is involved, adding an obstacle term, while leaving the drift and volatility terms untouched [2603.12375].

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 [2412.12213].

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 [2603.12375, 2412.12213].

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 $K$. Early-exercise features of American style require additional obstacle or enforcement terms [2603.12375]. In the option-pricing formulation, training stability is addressed by clipping $\Delta$ and $\Gamma$ within theoretical bounds, specifically $\Delta\in[0,1]$ and $\Gamma\ge 0$ [2412.12213]. 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.

Source: https://www.emergentmind.com/topics/finance-informed-neural-networks-finns