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No closed form formula but the $B(t, T)$ and thus $S(t)$ are functions of the spot rate $r_t$, and $r_t$ has a Gaussian distribution (details in e.g. "brigo mercurio interest rate models theory and practice") so building the distribution of $S(t, r_t)$ is straightforward.


By regressing $r_{i,t}$ on $y_{i,t}$ you are implying that: $$ r_{i,t} \equiv y_{i,t} - y_{i,t-1} = c_1 y_{i,t} + c_2 + \epsilon_{i,t}$$ This seems quite odd to me initially. If you assume that daily yield changes are independent with mean zero, then; $$ y_{i,t} = y_{i,t-1} + \xi_{i,t} \; \quad E[\xi_{i,t}]=0$$ Which can be replicated in the linear ...


Nominal rates have been negative in Europe for a while now. So the idea that rates should be non negative (the usual argument being that one would keep his money in his wallet rather than paying to lend it) is no longer a "first principle" of mathematical finance.


The confusion is that you think that we define the numeraire as this exponential function... It is not the case. We give the numeraire properties to $N$, then we model it. Similar to any other model. All we know is that $N$ is positive, and we have $$\frac{V_t}{N_t}=E^{N}\left[\frac{V_T}{N_T}|\mathbb{F_t}\right]$$ where $V_t$ is a tradable asset. $N$ can ...


I think you have a little misunderstanding. OIS just means the rate for fed funds. Usually people are referring to "FEDL01 Index" on Bloomberg. That's the VWAP of trades for the previous day in Fed Funds with participants lending to each-other. That's all in the past. That tells you nothing about the future. The Fed Funds futures settle to the average ...


This is rather similar to the solution you mentioned in your question :) Let $(r_t)$ be the short rate with $\int_0^{t}r_s\mathrm{d}s\sim N(0.03t,0.25t)$ and $B_t$ the value of the bank account. Recall that by definition $\mathrm{d}B_t=r_tB_t\mathrm{d}t$ and thus $B_t=B_0\exp\left(\int_0^t r_s\mathrm{d}s\right)$. Thus, $(B_t)$ is for every time point $t$ ...

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