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4

CDS quotes are observable. But none of: probabilities of default, hazard rates, loss given default/recovery, etc are observable. To get some kind of (risk-neutral) probabilities of default, many people make a lot of assumptions, in particular, that the hazard rate is constant (or if you're lucky enough to have CDS quotes at more than one tenor, then ...

1

I agree with @Kermittfrog's comment, that this only works if you do not impose any budget constraint (in the sense that your weights sum up to one). Other than that, I am sorry that I can not precisely answer your question where it was first derived (tbh: I am not even sure if it was explicity derived at all somewhere because it simply follows from the very ...

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For a payoff like this, it may be possible to replicate it by a linear combination of calls and puts. From there you can get the option's Vega by using the BS Vega of the calls and puts in your replication.

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There is no closed-form analytical solution for the long-only minimum-variance portfolio. Only the the unconstrained (short-sales allowed) portfolio. See here. Modifying the unconstrained portfolio to become the constrained portfolio in the manner you described is not going to be equal to the true constrained portfolio solution, which must be obtained by ...

3

$$\mathbf{E}\left[X+Y\right] = \mathbf{E}\left[X\right] + \mathbf{E}\left[Y\right]$$ This is just a property of random variables (see here).It doesn't matter that $X$ and $Y$ are not independent, or correlated. So yes, when you have a linear combination of derivatives, the value is the linear sum of the values of the individual derivatives.

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