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How often are closed-form equations actually useful for real world problem domains? When i did my PhD in applied math, they mostly came up in abstracted toy pro

How often are closed-form equations actually useful for real world problem domains? When i did my PhD in applied math, they mostly came up in abstracted toy problems. Then you get into the real world data or a need for realistic modeling and it's numerical methods everywhere.

danuker 4y ago
I find them most useful when there are many variables, or when I can see there's a relationship but I don't feel like trying out equation forms manually. It is indeed of limited use, since often I can spot the relationship visually. And once I get the general equation I can easily transform the data to get a linear regression.
chimeracoder 4y ago
> How often are closed-form equations actually useful for real world problem domains? When i did my PhD in applied math, they mostly came up in abstracted toy problems. Then you get into the real world data or a need for realistic modeling and it's numerical methods everywhere. And closed-form equations are themselves almost always simplified or abstracted models derived from real-world observations.
version_five 4y ago
Engineering?
mellavora 4y ago
Well, Black-Scholes has proved pretty useful. With the caveat that all models are wrong-- and most people using B-S know this. Which is why actual option prices have the "smile", with tail prices being higher than the model would predict (because traders know that the model underestimates tail risk, and generally have a good sense of how far it underestimates it, because the model is fairly transparent). Because B-S is closed form, you can run it backwards, to convert actual prices to an implied volatility. Which is also known to be wrong, because historical standard deviations of returns are only somewhat predictive of future observed returns. As one person put it, Black-Scholes is the wrong model, into which you put the wrong data, to get the right answer.

danuker 4y ago

I find them most useful when there are many variables, or when I can see there's a relationship but I don't feel like trying out equation forms manually. It is indeed of limited use, since often I can spot the relationship visually. And once I get the general equation I can easily transform the data to get a linear regression.

chimeracoder 4y ago

> How often are closed-form equations actually useful for real world problem domains? When i did my PhD in applied math, they mostly came up in abstracted toy problems. Then you get into the real world data or a need for realistic modeling and it's numerical methods everywhere. And closed-form equations are themselves almost always simplified or abstracted models derived from real-world observations.

version_five 4y ago

Engineering?

mellavora 4y ago

Well, Black-Scholes has proved pretty useful. With the caveat that all models are wrong-- and most people using B-S know this. Which is why actual option prices have the "smile", with tail prices being higher than the model would predict (because traders know that the model underestimates tail risk, and generally have a good sense of how far it underestimates it, because the model is fairly transparent). Because B-S is closed form, you can run it backwards, to convert actual prices to an implied volatility. Which is also known to be wrong, because historical standard deviations of returns are only somewhat predictive of future observed returns. As one person put it, Black-Scholes is the wrong model, into which you put the wrong data, to get the right answer.