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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!

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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".

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  • $\begingroup$ Upvoted -- I wonder what Bowen Chen will use the machine learning for once the F.D. schemes have already established the boundary. $\endgroup$ – Brian B Nov 9 '16 at 1:49
  • $\begingroup$ Good question. The feed-forward in an already calibrated neural network is likely to be significantly faster than finite difference schemes. Given good estimates of the boundary at a few points, you could then construct fast analytical price approximations by e.g. valuing the American option as a piecewise exponential barrier option with rebate. I am quite sceptical about the ability to accurately model the boundary for general dividend structures though. $\endgroup$ – LocalVolatility Nov 9 '16 at 1:55
  • $\begingroup$ Not sure I understand. If you need to run an FD scheme to calibrate a ML method which should allow you to generate exercise boundaries, for sure the latter won't be faster than the former, on which it relies. Of course one could object that the ML method is already calibrated when you use it, but in that case why not just store the boundary as part of the FD method. I'm probably missing out something. @Bowen Chen, could you better explain what you are trying to achieve? $\endgroup$ – Quantuple Nov 9 '16 at 9:55
  • $\begingroup$ Thank you for all of your answers! I would like to ultimately use machine learning to evaluate real option decision boundaries. Right now American Put only have one stochastic factor. I would like to train neural network on the single stochastic factor and see how well it performed. Eventually I would like to run the algorithm on multiple stochastic factors. Do you think this is something achievable? $\endgroup$ – Bowen Nov 9 '16 at 14:50
  • $\begingroup$ @Quantuple: I guess the idea is to once generate a relatively large training set of exercise boundary points at for random times-to-maturities and model parameters and then train the network to match these subject to some cost function. Given the trained network, you can then quickly compute the the exercise boundary through a feed forward. I think this could work well if you only consider continuous dividends. Apart from it being an academic exercise I don't really see the point yet either. I use a similar approach for getting initial estimates for implied volatility smile fitting problems. $\endgroup$ – LocalVolatility Nov 9 '16 at 20:00
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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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  • $\begingroup$ I just found out the same problem. The attempt seems superfluous. Do you think it is possible to let machine learning find boundary itself without generating the boundary? $\endgroup$ – Bowen Nov 10 '16 at 16:13
  • $\begingroup$ No, because there is no magic trick here you must learn from somewhere and this is not easy especially as @Local Volatility rightly points out: discrete cash dividends will create singularities in the exercise boundary. $\endgroup$ – Quantuple Nov 17 '16 at 21:02

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