I can see that might be the case, in an extremely deep, and yet strangely meaningless, way!Well, OK. By the same rationale, I guess one could say all of biology and chemistry is simply applying fancy techniques to the same boring equation (non-relativisitic Schrodinger equation).For a moment I thought these were good examples. Who doesn’t like an eikonal equation?! But really these just come from applying different techniques to the usual type of governing equation. I don’t think fancy techniques applied to boring equations quite satisfies me, I’m afraid!There is a fair amount of interesting math (at least to me) in my second volatility book. Some examples:
1. I took one course in general relativity as a senior at Caltech. But, I think I learned more Riemannian geometry in doing my chapter on "Advanced smile asymptotics" then I ever learned in that course. If non-linearity is your criterion for "interesting", you get that in the interplay between the eikonal and geodesic equations.
2. I have a chapter on "Spectral theory for jump-diffusions". This is a relatively undeveloped area -- as the finance operators are typically non-self-adjoint. (99% of textbook stuff is for self-adjoint operators). I came to appreciate some parts of functional analysis much more in writing this chapter.
3. Continuous-time inference for diffusions yields some interesting puzzles, esp. when boundary behavior is involved; for example, with slowly-reflecting diffusions or even just the square-root volatility process. I came to appreciate Girsanov theory more from writing that one. As part of that chapter, you may remember this nice thread (although many of the equations seem to be lost).
Delighted to -- thank you. Two amazon links:Alan, could you possibly post a link to your book?
You are probably speaking as a physicist, who are less concerned with (or maybe do not know) the mathematical niceties involved with PDEs. Alan's book is the most rigorous to date in the PDE literature galaxy. Most leave out the meat which leaves you high and dry when you try to implement it. Alan goes to the quick..Thanks! I have little experience with math finance, especially compared to you or Paul, but tend to side with Paul on this one (everything revolves around the same type of an equation). Still, such finance books I could actually read
Regarding your remark about biology and chemistry ("applying fancy techniques to the same boring equation (non-relativisitic Schrodinger equation)"), large parts of the disciplines require the Dirac equation, and when it becomes too complicated to solve, they experiment with other ways of modelling (e.g. statistical models/ML).
ML afficiandos claim they can solve 100-dimensional PDEs before breakfast.Any concrete suggestions for alternative maths or just grumbling?
By the way, nobody cares about derivatives pricing any more, another PDE for fixed income.... yawn!
At least if it were a different PDE...but it's the same one every time. (There is the nonlinear wave equation model...Epstein & Wilmott...that's more interesting!)Any concrete suggestions for alternative maths or just grumbling?
By the way, nobody cares about derivatives pricing any more, another PDE for fixed income.... yawn!