It seems like the problem of trying to estimate model parameters for continuous time models is not commonly covered in standard econometric textbooks, even those focusing on time series. I certainly am able to read and work on research pertaining to discrete time series models, but I have never encountered an introduction on how to estimate continuous time models using discretely sampled data and I would be curious to know if anyone has suggestions.
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$\begingroup$ You may like this old paper academia.edu/1903778/… $\endgroup$– Dimitri VulisOct 30, 2020 at 23:02
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This is seen as a bit of a niche field, which is likely why there are not so many books and these issues are not covered in standard econometrics texts. Options pricing models are usually fitted to options data rather than estimated econometrically from historical data. For statistical models, it is often more convenient to start from a discrete model as the data is discrete anyway.
However, there are some relevant books. You could check e.g. "High-Frequency Financial Econometrics" by Ait-Sahalia and Jacod, which covers the estimation of diffusion and jump models.
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1$\begingroup$ bergstrom is definitely the person when it comes to continuous time econometrics. But I can't say that I've ever tried to read his material. The references here or the paper itself might be useful. link.springer.com/chapter/10.1007/978-1-349-20865-4_5 $\endgroup$ Oct 31, 2020 at 14:17
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$\begingroup$ @markleeds Thanks for adding. I am not an expert on this field, but did some readings few years back. $\endgroup$– fesOct 31, 2020 at 14:38
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$\begingroup$ @fesman My experience with option pricing models has mostly been with GARCH models and, in this literature, people usually use a joint likelihood: a Gaussian likelihood in the implied volatility space plus whatever distribution is adequate in the time series space. Some people will prefer to use the time series only to estimate the dynamics under P and will use the IV in a second step to estimate the parameters of the pricing kernel (so, to convert to Q). But I am not used at all to continuous time models and their estimation. $\endgroup$– StéphaneOct 31, 2020 at 16:59
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$\begingroup$ Sure there are applications. Continuous time can be more convenient with some HF models. Also term structure models are often postulated in continuous time and estimated using historical time series. I might be wrong but my sense is that GARCH options pricing models are not widely used in practice. $\endgroup$– fesOct 31, 2020 at 20:13
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$\begingroup$ @fesman: I'm definitely no expert myself. I am just familiar with bergstrom from when I was looking at what happens when one starts aggregating time series ( say daily to weekly for example ). He's got a textbook that I have and, at a glance, it looks somewhat readable. Of course, the term readable is always in the eyes of the holder. :). $\endgroup$ Nov 1, 2020 at 13:39