Why Continuous-Time Models Remain Useful for American Option Pricing

Summary

The document asks why American options, which allow exercise at any time up to expiry, receive attention when actual trading and exercise occur discretely. It contrasts them with Bermudan options, where exercise is available only at specified dates, and notes that a daily exercise schedule can be priced through backward induction. In principle, refining the time grid approaches the American exercise problem, though very fine discretization can be difficult in practice.

The response frames continuous and discrete time mainly as choices of mathematical convenience rather than competing descriptions of reality. Continuous-time models can accommodate uneven time steps, aggregate naturally, and handle distributions that change over time. In practical valuation, however, models are discretized regardless, so the useful approach is to examine how results behave as the step size is reduced. The discussion offers no numerical comparison or pricing algorithm beyond backward induction; it is a conceptual explanation of why continuous-time formulations remain useful despite discrete market activity.

Key ideas

• American options allow exercise throughout the contract period, while Bermudan options restrict exercise to scheduled dates.
• A Bermudan option can be priced by backward induction over its exercise dates.
• Refining the time grid can approximate continuous exercise, although finer grids increase computational difficulty.
• Continuous-time formulations are convenient for uneven time steps, aggregation, and evolving distributions.
• Practical calculations discretize continuous-time models, so sensitivity to the time step is relevant.

Tags

# Why do we care about American options?


# Why do we care about American options?












I have been told most real options are American.

However, this isn't really true. Markets are closed at times, there are delays in transactions, or the owner of the option might be sleeping, or just otherwise not keeping up with the markets. So real life behavior is very discrete.

So why are we so much interested in pricing options where you can exercise at any point in $[0, T$]? Especially when that problem is very hard?

Is it not far more realistic to be interested in Bermudan options with e.g. daily-monitoring (252 days per year)?

Bermudan options can also be far more easily priced than Americans. Just use backwards induction with a stepsize $\Delta = 1/252$.

An American price would be obtained if the stepsize $\Delta \rightarrow 0$, but that is very hard to calculate in practice.

So if Bermudan options are more realistic and easier to price, why care about American options that seem mostly a unrealistic mathematical construct?

## Answer by Stéphane (score 1)

https://quant.stackexchange.com/a/51423

Working in discrete time or continuous time is mostly a matter of convenience. What most people do in some field of finance or economics is suggestive of what tends to be easier, though it's a kind of rule of thumb.

Off the top of my head, CT has the convenience of easily handling uneven time steps and allowing easy aggregation. It also handles changing distributions seemlessly, which is why a lot of progress in macroeconomics as of lately happened in CT (because you try to deal with an entire distribution of income, wealth, etc. and it changes over time).

Now, you could pick a very small $\Delta$ and see for yourself what happens when you follow your own advice. In practice, you'll discretize the damn thing anyways.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.