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I'm new to the stochastic calculus, and I keep converting the continuous stochastic differential equation to its counterpart in discrete time, such as the autoregressive models. I wonder in practice, what is the advantage of the continuous version over the discrete version? It seems to me the discrete version always has the advantage of being easier to simulate numerically and easier to understand, but I do not see the advantage of the continuous version.

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  • $\begingroup$ Most courses teach discrete before continuous because, as you point out, discrete analysis is easier to comprehend and simulate. Continuous methods are typically shorter to write out and easier to transform. $\endgroup$ – David Addison Jul 14 at 22:48
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You're right. Discrete models are easier to simulate and indeed, if you have a time continuous model, you typically first discritse it before you can implement it (computers live and work in a discrete world) (This, by the way, gives rise to discretisation errors). So, why do we love continuous models then? Because they're much easier. Think about the Black-Scholes model and its solution. A very easy formula, simple to derive and easy to generalise. The solution for its discrete counterpart, the Binomial tree from Cox-Ross-Rubinstein, is nastier to write down and work with. Even discounting with $e^{-rt}$ is more elegant than with $\frac{1}{(1+r)^t}$.

So really, we simply use time continuous models for the nice and elegant theory we know from stochastic calculus and Brownian motion. Discrete models are however rather useful for both, implementing the model and to obtain some intuition what's going on and what the model really does model.

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There are several reasons why continuous models are often preferred:

1.) It is more realistic: Trading happens continuously. Especially regarding HFT, trades can and might happen at any time.

2.) It therefore is also more coherent. You don't have to find and deal with artifical time steps.

3.) It is easier to handle. The lack of artificial time steps might produce more "closed" forms. In the classical black scholes model for example the only parameter to choose is the volatility. Considering a CRR model which is discrete, is way harder to calibrate.

When it comes to simulation you are obviously right, you often have to discretize. Not always though! Some models might have an analytic expression such as the Black-Scholes model. Then also most Operations within that model might have a closed form. Further all continous models live in the "same world" and are therefore easier to compare.

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