Choosing Continuous-Time and Discrete-Time Models in Finance

Summary

The discussion compares continuous-time finance models with discrete-time numerical approaches. One answer favors continuous time when it produces closed-form solutions, while suggesting discrete methods when a continuous model lacks one, since numerical grids over state variables can be easier to work with than stochastic differential equations. This frames the choice as a practical trade-off rather than a universal rule.

A second answer focuses on option pricing, arguing that continuous models can be a poor fit for early exercise and exotic features, and pointing to trees or Monte Carlo methods as alternatives for American options. The exchange offers conceptual guidance, not a systematic comparison or supporting performance evidence. Its claims are broad: it does not discuss particular model assumptions, calibration, computational costs, or cases where continuous-time methods can still handle early exercise numerically. The useful takeaway is to choose a framework according to the contract features and solution method required, and to distinguish analytical convenience from the ability to represent the product accurately.

Key ideas

• Continuous-time models can be attractive when they yield closed-form solutions.
• When closed-form solutions are unavailable, discrete numerical methods may be easier to implement.
• Early exercise and exotic features can complicate continuous-time option valuation.
• The discussion recommends trees or Monte Carlo methods for some American option problems, without comparing their accuracy systematically.

Tags

# When to use continuous time math vs discrete time?


# When to use continuous time math vs discrete time?












Seems that the theory books are all integrals in continuous time, yet in practice, discrete estimations works fine.

As a newbie to this, when do you choose to use the continuous time finance vs discrete estimations?

## Answer by phdstudent (score 4)

https://quant.stackexchange.com/a/40310

There is no hard rule. Usually continuous time models are good because they allow for closed form expressions for the solution (where discrete time models do not allow). However, if a given model in continuous time does not allow for a closed form solution then you are better off going to discrete time as it is easier to solve numerically using grids for state variables rather than solving numerically stochastic differential equations.

## Answer by Hui (score 0)

https://quant.stackexchange.com/a/40318

I tend to disagree with @phdstudent. In option pricing world, for example, continuous models are incapable to address early exercises on American options. Although there are some analytics models like Berjerksund-Stensland or Whaley model, which have closed-form formula, however, their accuracies are not as good as tree models or Monte Carlo simulation based model. Continuous models barely not work for exotic options. In my mind, continuous models only work well on European options.

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