Daily AI Trading Paper

Today we’re looking at a paper that sits right at the intersection of quantitative finance, stochastic control, and modern deep learning: using neural networks to solve a high-dimensional Black–Scholes equation when transaction costs are included. That matters because once you add realistic frictions to an otherwise elegant pricing model, the math becomes nonlinear, and the usual closed-form or low-dimensional numerical tools start to struggle. In practice, option desks care about this because transaction costs are not a minor detail. If you are continuously re-hedging a book, the cost of trading the hedge can materially affect the true value of the derivative and the quality of the hedge itself. To set the stage, the classic Black–Scholes–Merton framework gives a PDE for the price of a derivative written on one or more underlyings that follow geometric Brownian motion. In the idealized version, hedging is frictionless. That assumption is powerful mathematically, but it is not fully realistic. Leland’s transaction cost model is one of the simplest ways to bring in those frictions. The key idea is that every time the hedge changes, you pay a cost proportional to the amount traded. Once you do that, the PDE is no longer linear in the same way, because the transaction cost term depends on the sign of the curvature, meaning the second derivative of the option price with respect to the underlying. That is the crucial source of nonlinearity here. Why does dimensionality matter so much? In finance, once you move from a single underlying to a basket of assets, the state space grows quickly. Traditional grid-based finite-difference methods scale very poorly as the number of dimensions increases, which is the curse of dimensionality. This paper is trying to show that a neural-network-based stochastic method can handle that setting more gracefully, while still respecting the structure of the pricing equation. The broader mathematical tool they use is the FBSDE, the forward-backward stochastic differential equation representation of a parabolic PDE. In that framework, the PDE solution is represented by a coupled stochastic system: a forward process for the underlying assets and a backward process for the derivative value. The paper goes one step further and uses a 2BSDE, which adds an extra stochastic equation for the gradient of the PDE solution, and in effect gives access to the curvature as well. That extra layer is especially useful here because the transaction cost term depends on the curvature. So instead of only learning price and delta, the method also learns curvature information and its evolution over time. The core modeling move in this paper is to derive the explicit 2BSDE system for the Black–Scholes equation under Leland transaction costs. In the simplified version they use, the transaction cost contribution can be expressed through the diagonal curvature terms, which turns the PDE into a Black–Scholes–Barenblatt-type equation. The important conceptual point is that the local volatility term is effectively modified by a factor involving the sign of the curvature. That means the PDE’s coefficients are no longer fixed in the same way as in the standard Black–Scholes case; they now depend on the solution’s second derivatives. This is exactly the kind of feedback loop that makes the problem hard and makes a 2BSDE formulation attractive. Now, how do they solve it? They use a single neural network, specifically a ResNet with sine activations. The network is trained to learn both the option price function and the curvature matrix components, rather than using separate networks for each quantity. That is an important design choice. In principle, one could use automatic differentiation to compute first and second derivatives of the learned price function directly, but in high dimensions that becomes expensive and can be unstable. So instead, they let the network explicitly learn the curvature-related object through the 2BSDE dynamics, and then use automatic differentiation only where needed. The training procedure is built around simulated paths of the forward stochastic process. The forward asset dynamics are discretized with Euler–Maruyama. Then the backward quantities are updated along those paths, and the loss combines several pieces. One part encourages consistency across time steps for the learned price and its gradient. Another part is a shooting-style terminal loss: starting from the initial value, they propagate the estimated terminal value forward and penalize the mismatch with the actual payoff at maturity. They also include terminal penalties for the value, gradient, and curvature quantities. In other words, the network is not just fitting a static surface; it is being trained to satisfy the stochastic dynamics and the terminal payoff simultaneously. The architecture and training details matter here because this is a fairly delicate optimization problem. They use a ResNet with sine activations, which is a sensible choice when you want smooth higher-order derivatives. They train with Adam in two stages. First they run a learning-rate range test to pick a good learning rate, then they train for up to 40,000 iterations with early stopping, followed by a second stage at one-tenth the learning rate for up to 20,000 more iterations. The experiments use 50 time steps and batches of 100 simulated paths. They also note an implementation detail that is easy to miss but important: when learning the curvature head, they do not let gradients flow through certain terms in a way that would bias the curvature estimate. That suggests they were paying attention to the stability of the multi-objective loss. On the experimental side, they benchmark the method on geometric basket call options and butterfly spreads, in dimensions 10 and 100. Those are good test cases because geometric baskets have a structure that can reduce the effective complexity, so you can compare against finite-difference reference solutions, but they still create nontrivial challenges in the presence of sign changes in curvature and in scaling to many dimensions. The paper uses common parameter choices including a constant risk-free rate of 0.1, maturity of 1 year, and a re-hedging interval of 1/52. They also mention using identical parameters across dimensions in some cases, which helps isolate the effect of dimensionality. The headline result is that the network achieves a time-averaged RMSE of less than 0.1 in both the 10-dimensional and 100-dimensional settings. That is the main quantitative takeaway from the abstract, and it is meaningful because it suggests the method is not just working in a toy low-dimensional case. More importantly, the paper claims agreement with reference solutions while also capturing the effects of transaction costs. So the result is not only that the network can fit a price surface, but that it can learn the nonlinear deformation induced by hedging frictions. What should we actually conclude from that? First, the method appears to scale to dimensions where classical grid methods become awkward. Second, the joint learning of price and curvature seems to be a practical way to handle a PDE whose coefficients depend on the solution’s second derivatives. Third, the use of a single shared network is attractive from an engineering standpoint, because it avoids training multiple separate models for related quantities. At the same time, there are some important limitations and things to watch. The paper relies on a specific transaction cost model, namely Leland’s approximation, and also makes a simplification by keeping only diagonal curvature contributions. That makes the problem tractable, but it is not the most general market microstructure model. The benchmark products are also structured products with special symmetry, so while they are useful tests, they are not the same as pricing a fully generic exotic in a real multi-asset market. And as with most neural PDE solvers, performance depends on careful loss balancing, architecture choice, and training stability. The fact that they had to manage gradient flow carefully is a reminder that these methods can be sensitive. So the broader message is this: this paper shows that a physics-informed, stochastic-control-inspired neural network can solve a nonlinear high-dimensional option pricing problem that incorporates transaction costs and curvature-dependent dynamics. It is a strong demonstration of how 2BSDE methods can extend beyond plain Black–Scholes, and it points toward a practical route for learning prices, deltas, and curvature-related quantities in settings where classical methods become expensive. The next question, naturally, is how robust this approach is under more realistic cost models, more irregular payoffs, and less symmetric underlyings. That is where the field will likely push next.