In recent years, deep learning methods have emerged as a powerful alternative to traditional numerical approaches for solving both forward and inverse problems of partial differential equations (PDEs) and have been widely applied across various scientific and engineering domains [1–7]. Currently, various deep learning methods have been proposed and successfully applied to the study of PDEs. Representative approaches include the physics-informed neural networks (PINNs) [3], the deep operator networks [6], the deep Galerkin method [1], the local extreme learning machine [5], and the random feature method [7], among others. Among them, PINNs approximate the solutions of PDEs by training neural networks to minimize a composite loss function that integrates physical laws, initial/boundary conditions, and experimental or simulated data. Subsequently, researchers from various fields have introduced a series of improvements and extensions to the vanilla PINN formulation, all of which were aimed at enhancing its accuracy and robustness in order to tackle increasingly challenging problems arising in studying various PDEs, such as those with highly nonlinear, multi-scale, chaos or turbulence behavior [8–14]. Therefore, benefiting from the flexibility of physical laws and the diversity of data sources, PINNs and their various variants have been widely applied to solving various PDE models and addressing practical problems in science and engineering.

Rogue waves (also known as freak waves, extreme waves, giant waves or killer waves) are extremely rare, transient, and isolated large-amplitude waves whose most distinctive characteristic is a crest height at least twice that of the surrounding waves [15]. In 1965, Draper first proposed the concept of the rogue wave, which he named ‘Freak Ocean Wave' [16]. However, the existence of rogue waves lacked direct observational evidence for a long time, until January 1, 1995, when the Draupner oil platform recorded the first complete observation of a rogue wave, marking an important milestone in rogue wave research, and this wave is therefore also known as the ‘New Year Wave' [17]. To gain a deeper understanding of the formation mechanisms and dynamic characteristics of rogue waves, extensive theoretical analyses and experimental studies have been conducted across various disciplines. For decades, rogue wave theory has demonstrated significant research value in fields such as fluid dynamics [18], multistable system [19], nonlinear optics [20], Bose–Einstein condensates [21], plasma physics [22], and microwave transmission [23]. Additionally, rogue wave theory has gained widespread attention in financial markets, where researchers use mathematical models of financial rogue waves to explain extreme events such as stock market crashes, sharp exchange rate fluctuations, and systemic financial crises, revealing their intrinsic connection to nonlinear fluctuation mechanisms and providing theoretical support for understanding and predicting financial risks [24–27].

For decades, various option pricing models have been developed to more accurately describe the complex and dynamic features of financial markets and provide reasonable mathematical explanations, aiming to characterize the dynamics of financial assets and the market pricing mechanisms. In 1973, Black and Scholes pointed out that, under a series of idealized assumptions associated with a complete liquid market (i.e. the absence of transaction costs, taxes, and arbitrage opportunities, together with the assumption that stock prices follow a geometric Brownian motion), once options are correctly priced, profits could be made by creating a portfolio of long and short positions in both the options and the underlying asset, and they derived the theoretical valuation formula for options, known as the famous linear Black–Scholes (BS) equation [28]: in which V = V(t, S) is the market value of option, S is the price of the underlying stock, t denotes time, the domain is defined as 0 ≤ S < ∞,  0 ≤ t ≤ T, T is the expiry date. The σ is the volatility of stock price, and r ≥ 0 is the risk-free interest rate. In this model, the underlying stock (asset) price S is assumed to follow a geometric Brownian motion with the volatility σ and drift μ, satisfying the stochastic differential equation dS(t) = μS(t)dt + σS(t)dW(t) [29], where W(t) is the standard Wiener process. The solution of the linear BS equation indicates that any derivative and any asset can be perfectly replicated or perfectly hedged by a portfolio of other assets available in the market (see [30]). The introduction of the BS model laid the foundation for modern financial engineering, with empirical evidence showing that its theoretical option prices closely matched market observations under ideal conditions, leading to widespread application in real trading and becoming a key cornerstone for subsequent derivative pricing theories [31].

However, these assumptions of a complete liquid market often fail to hold in reality, as real markets are subject to factors such as transaction costs, insufficient liquidity, and volatility fluctuations. Consequently, in recent years, incomplete markets and markets characterized by limited liquidity (or illiquidity) have attracted increasing attention. Due to transaction costs, large investor preferences, and market incompleteness, classical BS models may give rise to strong or fully nonlinear dynamics behaviors, in which the volatility σ and drift μ can depend on the stock price S or the derivatives of the option price V itself. Therefore, as noted in [32–34], the most general equation governing European option prices is the following nonlinear BS equation: Here, $\tilde{\sigma }(S,{V}_{S},{V}_{SS})$ denotes the modified volatility function. In this work, we primarily focus on a transaction cost model with a wave function of the following form [35]: in which σ is the constant (historical) volatility, and ρ indicates a parameter modeling the liquidity of the market under study, we have σ, ρ > 0. Thus the nonlinear BS transaction cost model can be written as: In particular, if ρ = 0, the market is complete liquid (and the above equation reduces to the linear BS equation), whereas for large ρ, trades exert a substantial impact on transaction prices. Therefore, in this work, ρ is generally assumed to be small. If r = 0, it implies that there is no risk-free return in the market, i.e. no time value of money. In financial terms, this corresponds to a zero interest rate environment, where the discount factor equals 1 and future cash flows do not need to be discounted. The above equation (1.1) with r = 0 has also been investigated in [32–34, 36]. In [34], Patsiuk and Kovalenko employed symmetry reduction to investigate several exact solutions of the nonlinear BS transaction cost model, and applied these solutions to several boundary value problems.

The linear BS equation can be solved analytically to compute the prices of European options, and is applicable to both call and put options. A European call option is a contract that grants its holder the right to purchase a specified asset (referred to as the underlying asset S(t)) at a predetermined price (known as the strike or exercise price K) on a specified future date (called the expiry date T). If the holder chooses to exercise the right, the counterparty or seller is obliged to deliver the asset. Accordingly, the value of the option at maturity, known as the payoff function, is given by V(T, S) = (S − K)⁺. Conversely, a European put option grants the right to sell the underlying asset, with the payoff function given by V(T, S) = (K − S)⁺ [37]. The nonlinear BS model typically accounts for more complex market factors than the traditional linear BS model, such as transaction costs, market liquidity, and volatility variations. These models may more realistically reflect market conditions by incorporating nonlinear terms or stochastic elements, thereby facilitating the pricing of European options. Nonlinear BS models generally require numerical methods for their solution, as their equations are more complex and highly sensitive to changes in parameters and independent variables, making analytical solutions difficult to obtain [38]. Therefore, in this study, we aim to employ advanced deep learning methods to numerically predict the evolution and nonlinear dynamics of European option prices under a nonlinear BS transaction cost model in illiquid and incomplete markets.

Moreover, the assumption of constant volatility in classical stochastic differential equations fails to capture the volatility skew observed in real markets and cannot adequately explain the sharp fluctuations in option prices during economic crises [39]. Especially after the 1997 Asian financial crisis, it became increasingly evident that traditional stochastic models could not fully capture the complex dynamics of financial markets, leading to the development of various improved mathematical models aimed at providing a more accurate representation of the highly dynamic and evolving nature of financial markets [40, 41]. In 1997, Baaquie proposed a novel way to option pricing by applying path integral methods from quantum mechanics, effectively constructing a connection between the price of a stock option and the Schrödinger wave function [42]. In 2004, Haven further proposed that if the option price is regarded as a state function, it satisfies a specific potential function of the Schrödinger equation, and pointed out that the solution of the BS equation can be mapped to the linear Schrödinger equation [43]. In 2009, in order to construct a pricing model that could more accurately capture the nonlinear complexity of the market, Ivancevic proposed a novel nonlinear option pricing model, known as the Ivancevic option pricing model [44], as follows: here φ = φ(t, S) represents the option price wave function, the dispersion frequency coefficient σ is the volatility (which itself can be either a constant or a stochastic process), and the Landau coefficient β represents the adaptive market potential, which is essentially a nonlinear Schrödinger (NLS) equation. Notice that time t in equation (1.2) can be chosen to be negative since the option price wave function φ is invariant under the translation transformation t → t + T_i. Then, four types of analytical solutions of the NLS equation are provided in terms of Jacobi elliptic functions, all starting from de Broglie's plane-wave packet associated with the free quantum-mechanical particle [44]. The validity of the solution was verified through unsupervised Hebbian learning or the supervised Levenberg–Marquardt algorithm, and it was demonstrated that the solution is consistent with the BS model under certain conditions [44]. In 2010, Yan presented the concept of financial rogue waves based on the Ivancevic option pricing model and demonstrated that financial rogue waves can serve as a phenomenological description of extreme events in financial markets (such as financial crises/storms) and related fields [24]. Therefore, the Ivancevic option pricing model can describe potential extreme financial phenomena, where the option price φ depends on the stock price S, and S itself is a time-dependent stochastic variable, making it difficult to determine the actual data (including upper and lower boundary data) for stock and option prices over a future period. In this study, we are more interested in how to integrate the Ivancevic option pricing model with state of the art deep learning method to predict potential extreme financial phenomena (i.e. financial rogue waves) using only initial data at the current moment, thereby enhancing the capability of proactive analysis for market risk management and financial stability.

Due to the ‘appearing and disappearing without a trace' feature of rogue waves, predicting financial rogue waves using deep learning methods driven solely by initial data is a highly challenging task. In other words, the predictability of financial rogue waves largely depends on the accuracy of the initial data and is extremely sensitive to small variations in initial conditions. This sensitivity is particularly pronounced in complex problems such as highly nonlinear, multi-scale, chaos or turbulence behavior, making it difficult for traditional methods to effectively model and predict. Moreover, the nonlinear BS transaction cost model is essentially a variable-coefficient PDE. Due to its strong nonlinearity, the multiscale nature of the risk-free rate r under small-parameter conditions, and its sensitivity to changes in the independent variables, conventional numerical methods often struggle to solve it efficiently. Recently, Wang et al pointed out that when dealing with complex tasks that are highly sensitive to initial data (such as highly nonlinear, multi-scale, chaos or turbulence behavior problems), the existing PINN formulation fails to respect the inherent spatiotemporal causal structure of the physical system's evolution, which may ultimately lead the PINN model to converge to incorrect solutions [13]. To address this issue, they proposed the respecting causality PINN (RCPINN) by constructing a reformulation of the PINN loss function that more explicitly reflects the physical causal relationships during the model's training. Multiple numerical experiments demonstrated that this simple yet crucial modification significantly improved the model's prediction accuracy. To the best of our knowledge, some existing studies have applied deep learning methods to solve the (1 + 1) dimensional focusing NLS equation for rogue waves [14, 45, 46], as well as financial rogue waves in the Ivancevic option pricing model and the pricing of European options under the linear BS equation [47], but these studies have all been based on Dirichlet boundary conditions. However, in real, complex, and dynamic financial markets, the upper and lower boundary data for future time periods cannot be obtained, making the setting of the Dirichlet boundary condition obviously inconsistent with the actual conditions of financial markets. Therefore, in this work, we for the first time employ the RCPINN to predict the evolution of financial rogue waves described by the Ivancevic option pricing model, as well as predicting the evolution and complex dynamics behaviors of European option prices under the nonlinear BS transaction cost model. The goal is to enhance the predictive capability for market extreme volatility events and the pricing of complex European options based solely on initial data, and to further explore the potential value of deep learning methods in financial risk management and option pricing theory.

The structure of this paper is as follows: section 2 provides a detailed introduction to the RCPINN and other enhanced techniques. Section 3 presents numerical results for predicting financial rogue waves and predicting the evolution of European option prices by employing the RCPINN only with initial data. Section 4 concludes the paper and discusses the future applications and developments of deep learning method in the financial market and financial derivatives.

In this part, we take a detail overview of the RCPINN [13] in the context of solving nonlinear option pricing models. For convenience, we take the prediction of financial rogue waves in the Ivancevic option pricing model as an example to detail the principle and architecture of the RCPINN.

• Respecting causality physics-informed neural network

Specifically, we consider the Ivancevic option pricing model with corresponding initial and periodic boundary conditions take the form where T_i and T_f respectively indicate the initial time and final time, while S_l and S_r indicate the left and right boundaries of space. ${ \mathcal I }$ is initial operator. For the sake of simplicity in neural network training, we set $\phi (t,S)=u(t,S)\,+{\rm{i}}v(t,S),\,{ \mathcal I }(S)={{ \mathcal I }}_{u}(S)+{\rm{i}}{{ \mathcal I }}_{v}(S)$ in equation (2.1), thereby u(t, S),  v(t, S) and ${{ \mathcal I }}_{u}(S),\,{{ \mathcal I }}_{v}(S)$ are real-valued function for neural network training. Subject to the initial conditions and periodic boundary conditions where φ describes the unknown latent solution which is governed by the Ivancevic option pricing model (2.1). Then, we represent the unknown solution φ(t, S) by a deep fully-connected neural network φ_θ(t, S) with $\tanh $ activation function, in which θ denotes all tunable parameters of the neural network (including weights and biases). Now, a neural network model can be trained by minimizing the following total loss function

Since we are considering the periodic boundary conditions (2.3) of the Ivancevic option pricing model (2.1), in order to further simplify the training objective (2.4), we rigorously enforce the periodic boundary conditions by embedding the input coordinates into Fourier expansion utilizing equation (2.12), with non-negative integer m. Then the total loss function (2.4) can be reduced to where hyper-parameters {λ_ic, λ_r} provide the flexibility to assign distinct learning rates to each loss term, enabling a balanced interaction among them throughout the model training process. The initial condition loss ${{ \mathcal L }}_{ic}({\boldsymbol{\theta }})$ can be defined as For respecting causality for training RCPINN, we suppose ${\{{t}_{i}\}}_{i=1}^{{N}_{t}}$ uniformly discretize the temporal domain [T_i, T_f] (here ${T}_{i}={t}_{1}\lt {t}_{2}\lt \cdots \lt {t}_{{N}_{t}}={T}_{f}$), and ${\{{S}_{j}\}}_{j=1}^{{N}_{x}}$ uniformly discretize the spatial domian [S_l, S_r]. Then the weighted residual loss ${{ \mathcal L }}_{r}({\boldsymbol{\theta }})$ can be defined as in which w_i should be large, thus allow the minimization of ${{ \mathcal L }}_{r}({t}_{i},{\boldsymbol{\theta }})$ only if all residuals ${\{{{ \mathcal L }}_{r}({t}_{k},{\boldsymbol{\theta }})\}}_{s=1}^{i-1}$ before t_i are minimized properly, and vice versa. the weights w_i can be represented as where ε is the causality parameter that controls the steepness of the weights w_i. Then, the residual loss ${{ \mathcal L }}_{r}({t}_{i},{\boldsymbol{\theta }})$ in equation (2.7) can be rewritten as

Clearly, the weights w_i exhibits an exponential inverse relationship with the magnitude of the accumulative residual loss from the previous time steps. As a result, ${{ \mathcal L }}_{r}({t}_{i},{\boldsymbol{\theta }})$ cannot be minimized unless all previous residuals ${\{{{ \mathcal L }}_{r}({t}_{l},{\boldsymbol{\theta }})\}}_{l=1}^{i-1}$ are reduced to a sufficiently small value, ensuring that w_i becomes large enough. This straightforward yet highly effective strategy ensures that RCPINN explicitly respects physical causality during the training phase, making it particularly well-suited for predicting financial rogue waves, which heavily rely on initial data.

The choice of the causality parameter ε. We must emphasize that the results generated by the proposed weighted residual loss ${{ \mathcal L }}_{r}({\boldsymbol{\theta }})$ are notably sensitive to variations in the causality parameter ε. One can observe that opting for a larger ε value might result in a more difficult optimization process, since the residuals at earlier times need to be minimized to extremely small values to trigger the latter temporal weights. Conversely, choosing a very small ε can impede the network's capacity to adequately minimize the residuals at subsequent times. Thereby, in order to choose an appropriate causality parameter ε and avoid laborious hyper-parameter tuning, we employ an annealing strategy by utilizing an incrementally increasing sequence of values ${\{{\epsilon }_{i}\}}_{s=1}^{p}$ that progressively strengthen the enforcement of the PDE residual constraints.

A stopping criterion for assessing training convergence. By monitoring the magnitude of the residual weights {w_i}, one can establish an effective stopping criterion to evaluate the convergence of the RCPINN model throughout the training process. Specifically, one can choose to stop training when $\mathop{\min }\limits_{i}{w}_{i}\gt \delta $, where δ ∈ (0, 1) is a selected threshold parameter.

• Additional enhancement technologies

In this part, we employ a few important extensions and necessary settings that can further enhance RCPINN's performance in terms of accuracy and computational efficiency.

Modified multi-layer perceptrons. From [48], Wang et al presented a novel neural network architecture by introducing two transformer networks that project the inputs variables to a high-dimensional feature space and utilizing a pointwise multiplication operation to update the hidden layers, whose forward propagation rule as where X indicates an batch of the input data points, σ represents nonlinear activation function, and ⊙ denotes a point-wise multiplication. Here we refer to this novel architecture as modified multi-layer perceptrons. All trainable parameters can be written as The modified multi-layer perceptrons architecture demonstrates greater effectiveness compared to standard multi-layer perceptrons in minimizing PDE residuals and accurately capturing sharp gradients.

Exact imposition of periodic boundary conditions. It is widely known that the inexact imposition of boundary conditions may harm the neural network's performance and training stability. For accurately imposing periodic boundary conditions, Dong et al proposed that periodic boundary conditions can be strictly enforced as hard constraints by constructing Fourier feature embedding of the input data [49]. Specifically, if we consider a smooth function f(t, S) with periodic P (namely f(t, S) = f(t, S + P)), the periodic constraint of f(t, S) can be encoded in a neural network by means of a one-dimensional Fourier feature embedding where $\omega =\frac{2\pi }{P}$, and non-negative integer m can be set arbitrarily. Thereby, for any network output F_θ, it can be proved that any ${{\boldsymbol{F}}}_{{\boldsymbol{\theta }}}({f}^{{\prime} }(t,S))$ exactly satisfies the periodic constraint.

Figure 1 displays schematic architecture of the RCPINN for predicting financial rogue waves of Ivancevic option pricing model. Figure 1 comprises four key components: (I). The network input with Fourier feature embedding is processed through a modified multi-layer perceptrons containing two transformation networks to generate outputs, where the Fourier embedding ensures exact imposition of periodic boundary conditions. (II). The Ivancevic option pricing model and its initial conditions (providing initial training points) are incorporated as physics-informed constraints, which connect to the network outputs from Part (I) via automatic differentiation. (III). The loss function ${ \mathcal L }$ structure explicitly illustrates the causal parameters ε, weights w_i of residual loss, initial condition loss ${{ \mathcal L }}_{ic}$ and residual loss ${{ \mathcal L }}_{r}$. (IV). The spatiotemporal domain [T_i, T_f] × [S_l, S_r] is uniformly divided (collocation points sampled via uniform grids), and Adam optimizer is utilized for network training, while a reconstructed residual loss function ${ \mathcal L }$ in Part (III) base on uniformly segmented time regions enforcing network respect the temporal causal structure during training.

In this part, we employ the RCPINN to predict financial rogue waves of Ivancevic option pricing model and providing the price of European option in nonlinear BS transaction-cost model. Here, we first present the corresponding hyper-parameters settings in this work. We set λ_ic = 100,  λ_r = 1, N_t = 50, N_S = 64. The incrementally increasing causality parameter sequence ${\{{\epsilon }_{i}\}}_{s=1}^{5}=[1{0}^{-3},1{0}^{-2},1{0}^{-1},1{0}^{0},1{0}^{1}]$, the threshold of stopping criterion δ = 0.99. The network architecture of RCPINN is configured as [2, 40, 40, 40, 2/1]. All networks are trained via stochastic gradient descent utilizing the Adam optimizer with default settings [50] and an exponential learning rate decay with a decay-rate of 0.9 per 5000 training iterations. All numerical experiments are done with 13th Gen Inter(R) Core(TM) i9-13900HX CPU processor and NVIDIA GeForce RTX 4060 GPU processor on the Windows 11 operating system.

This study addresses practical needs in financial markets and option pricing theory by innovatively applying the state of the art RCPINN to predict extreme rogue wave phenomena and European option prices in financial markets. We select the Ivancevic option pricing model (for describing potential extreme financial phenomena) and nonlinear BS transaction cost model (for pricing European option prices) as our research model. Due to the inherent uncertainty of stock prices in real financial markets, obtaining actual future data for both stock prices and option prices remains particularly challenging. Therefore, unlike prior studies, this work presents the first successful prediction of extreme financial rogue waves and European option price evolution using the Ivancevic option pricing model and the nonlinear BS transaction cost model based solely on limited initial data, highlighting the potential of deep learning in financial risk management and option pricing theory. We also provide a comprehensive introduce of the RCPINN, by rigorously preserving the inherent spatiotemporal causal structure of physical system evolution, this deep learning method proves particularly effective for investigating complex tasks exhibiting high sensitivity to initial data, including strongly nonlinear, multi-scale, chaotic, or turbulent behaviors. Then, the schematic architecture of the RCPINN is exhibited, which comprehensively demonstrate both the core methodology and enhanced technical features.

Financial rogue waves represent precisely this type of complex nonlinear localized wave phenomenon, characterized by their abrupt emergence and disappearance, making them notoriously difficult to detect and predict. Under given initial conditions of the Ivancevic option pricing model, our RCPINN framework successfully predicts two canonical types of financial rogue waves (namely first-order and second-order financial rogue waves) at specified future time points t and stock prices S. Moreover, the nonlinear BS model poses numerical challenges due to its strong nonlinearity, the multiscale nature of the small parameter r, and the sensitivity of its variable coefficients to changes in the independent variables. Based on the nonlinear BS transaction cost model, we employed the RCPINN to successfully predict the price evolution and dynamic characteristics of both European call and put options, offering a novel approach to European option pricing. This study not only provides a novel research tool for predicting extreme rogue wave phenomena in financial markets and related fields, but also offers an effective method for the numerical prediction of European option price evolution and its nonlinear dynamics. Furthermore, it presents a new technical approach for extreme financial risk early-warning systems and holds significant practical value in applications such as forecasting European option price evolution, exchange rate fluctuations, liquidity crisis monitoring, market crash prevention, and derivative pricing.