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I need to calibrate Vasicek's model $dr_{t} = a(\theta - r_{t})dt + \sigma dW_{t}$ in a market with no swaptions. I was thinking in estimating $\sigma$ with historic data, but I'm in the doubt with calibration for the rest of the parameters (to term structure). I thought of two ways:

  1. Minimize $\sum_{i}(P^{market}(t, T_{i})-P^{vasicek}(t,T_{i}))$
    • Where $P^{vasicek}(t,T_{i})=e^{A(t,T_i)-B(t,T_i)r_t}$. $A$, and $B$ are Vasicek known formulas depending on $a$, $\theta$ and $\sigma$.
  2. Minimize $\sum_{i}(r^{market}_{t_i}-r^{vasicek})$
    • Where $r^{vasicek}_{t_i}$ is the average of the short rates obtained by Vasicek dynamics in time $t$

Are both approaches equivalent? Is one better than the other? Is there a better way?

PD: $r^{vasicek}$ is missing the subscript $t_i$ because for some reason if I add it LaTeX does not compile in that line.

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    $\begingroup$ Would it be possible to draw some correlations between the rates with no swaptions and markets with swaptions and then use some transformation of the markets with swaptions to the market without? Thats how some less liquid currencies tend to be generally conceived. $\endgroup$
    – Attack68
    Nov 13, 2023 at 19:51

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