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I have been reading the paper "Bridging P-Q Modeling Divide with Factor HJM Modeling Framework" by Lyashenko and Goncharov (2022). On Equation 5 of page 4 of the paper, I came across the volatility process $\Sigma_{t}$ which is a K x N matrix where K is the dimension of basis vector and N is the dimension of Brownian motion. Assuming I have the data for spot rates at many maturities and at every time, how can I find this volatility process term?

I know that it can't be the same as volatility of spot rate or the forward rate. For example, given the screenshot below from page 18 of the paper, I have $R_{t}$ and can find out $B_{R}$. The issue is to find out the volatility processes $\Sigma_{t}$ or $\Sigma_{t}^{R}$.


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1 Answer 1

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$\boldsymbol{\Sigma}_t$ is a $K \times N$ matrix volatility process for the factor vector $X_t$, and $dX_t = \ldots dt + \boldsymbol{\Sigma}_t dW_t$.

See equations (19) and (20)

enter image description here

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