For your question, definitely! Whenever you have a matrix, there must be some use for the eigenvalues and eigenvectors.

When we have 2 unknowns and 1 constraint, we showed how to reduce one of the variables to get a standard optimization problem for a single variable like for calculus.

When you have 3 unknowns and 2 constraints you can do the same thing. But for any more variables than that, the best you can do is to reduce the number of variables from n to n-2 by using the constraint equations. That still leaves an (n-2)-times-(n-2) matrix in the new variables. Then you would want to minimize that which you would do by finding the smallest eigenvalue.

]]>We will not talk at all about that in class! This is truly unfortunate. But the fact is that in the very simple scenarios we study in class, we always assume that the fluctuations are *continuous* in time, in fact we even assume that sigma is always finite (for simplicity we assume it is constant) which means that the variance has a derivative. That does not mean that the stock price is differentiable. It is not. But we do assume the stock prices are continuous. In fact, in worst case scenarios stock prices do jump discontinuously.

I would refer to these as “nonlinear” effects because of an analogy to some ideas in physics. In a first course in physics you focus on purely elastic materials using springs and masses and partial differential equations like the wave equation. But you know that there are inelastic phenomena, like bending a metal until it breaks. That is what happens in bad scenarios in finance. I think that real human consideration has to be used for that. Not just automatic mathematical formulas. There are some things people are better at (than computers).

]]>For the Poisson versus binomial the basic rule is this: the binomial distribution has two parameters n, the number of trials, and p, the probability of success for each trial. If you take the limit where n goes to infinity, and p goes to zero, in such a way that the product

np converges to a limit lambda, then you will get a Poisson variable with parameter lambda.

So if n is large and p is small, you can approximate a binomial (n,p) random variable by a Poisson random variable with parameter lambda, where lambda is calculated as lambda=np.

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