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If you want to address interesting problems that are interesting for financial mathematics, I do not believe you have the good list. Pricing. For instance, most of explicit formulas for pricing that are not available yet will never be. In this direction, you should have a look at simulation techniques. See for instance Nonlinear Option Pricing. Interesting ...


Just read some papers recently. Hope it is helpful. Some model I know with such Bang phenomena is called: Jump-Diffusion Model. The idea is a little different. We add Poisson counter term into the equation. On the other side, there is lots of work on test of such Bang phenomena, which is called "structural break". One early reference is the following: ...

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