what's the best black scholes tutorial?
The "Natenberg Bible": https://www.amazon.com/Option-Volatility-Pricing-Strategies-...
PDEs you should know
21–30 of 102 posts
Re: PDEs you should know
#22It's been awhile since I've had a Diff EQ class, but isn't the harmonic motion one an ODE?
Yeah good catch
Re: PDEs you should know
#23PDEs are really useful if you are in the rare domains where they are useful. But most PDEs don't have even have closed form solutions for non-trivial boundary conditions. So unless you are a physicist or something adjacent, no, no you really don't need to know these. Hard to say what's the intended audience for the page though. Could be a message aimed at physics undergrads or something. If so, then indeed, you shoul…
As an engineer, I've used lots of FEA to solve problems, but if you can't put part of your solution space where it can be approximated by a closed form solution (and there are many more than those shown, which are amenable to solution to appropriate methods), you're going to have a hard time building a trusted model. There's a reason we still have wind tunnels.
The most interesting parts of good FEA (where you've shown your model and reality match on measurable components), is that you can see hidden and unmeasurable variables, which may be design limiting.
Re: PDEs you should know
#24Earlier quoted context omitted.
Why does a PDE need an analytical solution? Are you arguing that e.g. Navier-Stokes isn't useful? edit: Just noticed that Navier-Stokes isn't even on here. This is frankly a weird list.
Well, to be useful, wouldn't you need to be able to use it? Most people outside of physicists and physics-adjacent fields are very far away from the mathematical tools to deal with these equations.
Numerical methods exist. There's a whole field to simulate PDEs that you can't solve exactly.
Re: PDEs you should know
#25Earlier quoted context omitted.
Most useful equations don't have a closed form solution. This is why things like machine learning exist. More specifically PDEs are a hot topic in DNNs at the moment. See physics informed neural networks: https://en.m.wikipedia.org/wiki/Physics-informed_neural_netw...
The point is that equations without closed form solutions are pretty useless if you aren't into pretty hairy maths, Green's functions and the like. An equation everyone should know is Hooke's law. That's useful at a high school level.
I think we need a better mutual definition of useful. IMO an equation is "useful" if someone out there is doing practical things with it. You are taking a sort of definition where anyone can do math with it.
Re: PDEs you should know
#26Earlier quoted context omitted.
Most useful equations don't have a closed form solution. This is why things like machine learning exist. More specifically PDEs are a hot topic in DNNs at the moment. See physics informed neural networks: https://en.m.wikipedia.org/wiki/Physics-informed_neural_netw...
The point is that equations without closed form solutions are pretty useless if you aren't into pretty hairy maths, Green's functions and the like. An equation everyone should know is Hooke's law. That's useful at a high school level.
The math isn't that hairy. I took numerical methods and computational physics classes in undergrad.
Re: PDEs you should know
#27PDEs are really useful if you are in the rare domains where they are useful. But most PDEs don't have even have closed form solutions for non-trivial boundary conditions. So unless you are a physicist or something adjacent, no, no you really don't need to know these. Hard to say what's the intended audience for the page though. Could be a message aimed at physics undergrads or something. If so, then indeed, you shoul…
> PDEs are really useful if you are in the rare domains where they are useful.
Aside from ‘rare’, this seems at best vacuously true.
> But most PDEs don't have even have closed form solutions for non-trivial boundary conditions. So unless you are a physicist or something adjacent, no, no you really don't need to know these.
As others have said, while your first sentence is surely true, the latter doesn't follow from it (and I would argue isn't true—but it depends on how you define adjacency). There are lots of things one can usefully do with an equation besides finding a closed-form solution. (For an ODE example, the classical predator–prey model does not have a nice closed-form solution, but is still plenty useful.)
Re: PDEs you should know
#28Who is "you" in this scenario? Who actually needs to know these by heart day to day?
Re: PDEs you should know
#29I think it’s better to know that sometimes we only know how to describe something by relating rates of change to other states. And that’s ok. Maybe it has a closed form equation, or maybe can only be solved numerically. But if I see that a differential equation looks like a wave equation, then I get intuition that it’s describing waves. And why do the waves appear? Because the physical process the PDE describes has a speed limit on information passing from time into space!
Don’t like traffic waves? Well, why is there some limit on spatial information connected to temporal information? It’s because I cannot see through the cars in front of me. The “fog of war” creates the waves. The denser the fog (e.g. I’m surrounded by semitrucks), the greater the likelihood of waves developing.
This intuition is formed by being able to recognize the form of the PDE with general knowledge of the solutions, without needing to actually solve the PDE. Sure, additional insights are possible if you solve it, but knowing that traffic is like springs gives you leverage to use your ordinary intuition to understand unfamiliar things.
Point of fact, James Maxwell of E&M fame saw the wave equation and the separate electric and magnetic field PDEs and came up with a detailed spring model to give himself a more familiar analog to play with.
Re: PDEs you should know
#30Earlier quoted context omitted.
Why does a PDE need an analytical solution? Are you arguing that e.g. Navier-Stokes isn't useful? edit: Just noticed that Navier-Stokes isn't even on here. This is frankly a weird list.
Well, to be useful, wouldn't you need to be able to use it? Most people outside of physicists and physics-adjacent fields are very far away from the mathematical tools to deal with these equations.
I think what the other commenters are getting at is that PDEs can be used without having a closed form solution (and mostly are used that way as closed form solutions usually only come up in special artificial cases). You start your system in a real known state and then propagate it forward in time using (for example) the finite difference method on the equations to figure out the state at a later time.
https://en.wikipedia.org/wiki/Numerical_methods_for_partial_...