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Yes. The map $R(\cdot;S,T):\mathbb{R}^{2}\to\mathbb{R}$ completely describes the forward rate/spot rate term interest rate structure for each $t\geq0$. (You can think of it as the market interest rate surface for the rate $R$ at time $t$). The notation $R(t;S,T)$ is meant to remind you that $R$ is a stochastic process for $t>0$, the periods of time ...

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I am note $100\%$ sure that I understand the question. But yes. More formally one could write $R(t,S,T)$ for the rate from $S$ to $T$ observed at $t$ and $R(t,t,T)$ for the spot.

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The negative solution does not satisfy $P(T,T)=P(t,t)=1$

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You need to take in to account the variation in return ( i.e. standard deviation) of the returns to get the VaR. Alternatively,you can simulate the total return using Excel as well and find the VaR accordingly.

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The central theme of Asset Pricing is in detecting what drives the evolution of the (stochastic) discount factor. Academic finance assume as a methodological starting point that expectations are rational, i.e. unbiased, hence when prices ( $p_t$ ) are determined as the expected value of the discounted (by the discount factor $M_{t+1}$) payoff of the security ...

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Your question depends on the discount factor you wish to use for pricing. If u use the risk-free rate (from the bond), it wouldn't be in line with the no-abitrage condition to assume an risk neutral agent can't/wouldn't invest in bonds to carry money into next period. To understand this: just assume a 1 period model with two outcomes for S, where both ...

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