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Regarding your first question. You can try to argue in the following manner. Using the fact that $E_{Q^t}[P(t,T)|\mathcal{F}_s]P(s,t)=P(s,T)$, then: $$p(s)=\lim_{T\to\infty}P(s,T)^{\frac{1}{T}}=\lim_{T\to\infty}(P(s,t)E_{Q^t}[P(t,T)|\mathcal{F}_s])^{\frac{1}{T}}=\lim_{T\to\infty}E_{Q^t}[P(t,T)|\mathcal{F}_s]^{\frac{1}{T}}$$ as ...

Several ways, some are applicable at certain times when others are not, sometimes you can chose freely which way to go. a) You can attempt to solve using partial differential equations b) If applicable you can find a discretization of the underlying model and run a monte carlo simulation c) you can setup binary trees and work your way backward. Here is ...

Regarding your first question, let me reformulate $p(s)$: $p(s)=(\frac{P(s,T)}{P(s,t)})^\frac{1}{T}= \frac{e^{-\frac{(T-s)R(s,T)}{T}}}{e^{⁻\frac{(t-s)R(s,t)}{T}}}.$ Now, because $R(s,t)$ is finite, the term $e^{⁻\frac{t-s}{T}R(s,t)}$ converges to 1 for $T\to\infty$. On the other hand, the term $e^-\frac{(T-s)R(s,T)}{T}$ converges to ...

It used to be (and maybe still is) Standard Securities Calculation Methods ..., a 2 volume set, but it is no longer available. You might find it in a library. It is available for $$$ at http://www.sifma.org/research/bookstore.aspx, or maybe you can find a used copy. TIPS, Inc., http://www.tipsinc.com/prods.htm, has an app and software implementing the books. ...

Increased volatility towards the event start is definitely from increased order flow. There are some papers specifically on "prediction markets", the ones with practical applications are on market making which I suspect is generally a loss-making operation conducted by the exchanges themselves when a market is opened. Given the short-time periods and small ...