A Fully Forward Deep-Learning Method Prices Compound and Bermudan Options

Pricing an option gets messy fast when the payoff depends on several future decisions or many underlying assets. This paper tackles that problem with a fully forward deep-learning method called the Compound BSDE method. A backward stochastic differential equation, or BSDE, rewrites pricing as a system that works backward from the final payoff to today. The authors extend the classical deep BSDE method, which handled one BSDE, to compound BSDEs. That lets them treat compound options and optimal stopping problems such as Bermudan option pricing in a single framework. They also establish convergence properties for the algorithm and derive an a posteriori error estimate, a way to assess the error after the computation is done. In numerical experiments, the method is reported to be accurate and computationally efficient. The paper says it is effective for high-dimensional option pricing and optimal stopping problems, which are the cases that usually become hardest for traditional numerical tools.

A Bermudan option gives you a few set dates to act. A compound option stacks one option on top of another. Each added choice makes the price harder to pin down. The Compound BSDE method does something odd and useful. It prices these contracts by marching forward in time. That is the surprise. Most pricing stories start at the payoff and work backward. This method keeps the clock running forward and still reaches the right answer. For traders, that matters when many future choices and many assets make old tools slow or awkward. For anyone who has waited on a yes-or-no choice, the logic is familiar. Timing changes value.

When one option sits on another

The method rewrites pricing as a system of backward stochastic differential equations, or BSDEs. A BSDE is a rule that starts from the final payoff and works back to today. That shift gives compound options a cleaner shape. It also fits optimal stopping, which means choosing the best time to stop. Bermudan pricing fits that pattern because exercise only happens on set dates. The new algorithm extends the older deep BSDE method, which handled one BSDE. It then adds a convergence result. It also gives an a posteriori error estimate, a check of the error after the run ends. Numerical tests showed accuracy and speed in high dimensions, meaning many assets or risk factors.

How a forward run can still solve it

The setup starts with simulated price paths. A neural network, meaning a pattern-finding model made of linked layers, learns the missing pieces. The run moves forward in time. At each step, the method updates the linked BSDEs, or backward equations. That lets one forward setup cover several layers of payoff. The price guess changes until the final payoff lines up. The convergence proof shows the scheme does not drift without control. The error estimate then checks the result after the run ends.

Why it matters for real pricing

Pricing gets hard when contracts depend on many assets and many future choices. Old methods can slow down or get awkward in that setting. The Compound BSDE method gives one forward route for both compound and Bermudan problems. That matters because one tool can cover high-dimensional cases that strain classic solvers. The error estimate also gives a built-in sanity check after the run.

The Bermudan geometric basket put option

The surprise here is not just speed. It is that a forward method can live inside a backward pricing world. That opens a practical path for compound options and Bermudan payoffs. This method's own tests point to the Bermudan geometric basket put option as a hard case. The next check is whether the same forward frame keeps working as dimensions climb. If it does, one of finance's most tangled pricing jobs gets a simpler route.