Finding an American Put Exercise Boundary with Trees or Simulation
Summary
The document discusses methods for estimating the early exercise boundary of an American put under geometric Brownian motion. For continuous or proportional discrete dividends, a binomial tree can approximate the boundary; fixed discrete dividends may be handled with a finite-difference method. Both approaches require a sufficiently fine spot grid for an accurate boundary estimate.
For a path-based approach, the answer recommends Longstaff–Schwartz least-squares Monte Carlo. Working backward through simulated paths, the method estimates continuation value by regression and compares it with the option’s intrinsic value to identify when exercise is optimal. The regression stage could also use supervised learning methods. A key limitation is that standard Longstaff–Schwartz produces a suboptimal exercise strategy and low-biased prices. The responses describe approaches, but provide no numerical accuracy or speed comparison.
Key ideas
- Binomial trees can approximate the exercise boundary for continuous or proportional discrete dividends.
- Finite differences are suggested when fixed discrete dividends are included.
- A fine spot grid is needed for accurate tree or finite-difference boundary estimates.
- Longstaff–Schwartz estimates continuation values by regression on simulated paths and compares them with intrinsic value.
- Standard Longstaff–Schwartz can produce suboptimal exercise decisions and low-biased option prices.
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Full text
# Use machine learning to find exercise boundary of American put option # Use machine learning to find exercise boundary of American put option I am working on using machine learning to obtain American Put's early exercise boundary. To train the model, I need an output label (known boundaries values). Is there a fast way to obtain the exercise boundary base on the simulated paths of Geometric Brownian Motion? Any inputs are much appreciated! ## Answer by LocalVolatility (score 2) https://quant.stackexchange.com/a/30927 From what I understand is that you are looking to approximate the exercise boundary under geometric Brownian motion dynamics. If you only consider either continuous and/or proportional discrete dividends, then the simplest approach would probably be to use a binomial tree. If you are interested in including fixed discrete dividends, then I'd recommend using a finite difference scheme. In both cases, you'd need to ensure that your spot grid is fine enough in order for your exercise boundary approximation to be sufficiently accurate. A good introductory reference for both methods is "Paul Wilmott on Quantitative Finance". ## Answer by Quantuple (score 1) https://quant.stackexchange.com/a/30930 The lattice/FD approach suggested by @LocalVolatility will work fine. However, since you specifically mention "based on the simulated paths of a Geometric Brownian motion", you could alternatively consider least-squares Monte-Carlo. More specifically, working backwards in time from the expiry to the inception of the contract, the Longstaff-Schwartz algorithm will allow you to work out continuation values from the simulated paths -- under the hood this conditional expectation is evaluated in the least squares sense using standard regression techniques (hence the original name of the method) but you could also use more elaborate supervised learning methods at this point I guess. The intersection of the continuation value and the intrinsic value for each time $t$ then by definition constitutes the optimal exercise boundary. The tricky part is that the standard Longstaff-Schwartz algorithm actually identifies a sub-optimal exercise strategy (resulting prices will be low-biased). On the sunny side, general dividend structures can be handled almost seamlessly in Monte Carlo.
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