Sensitivity of prices and yields to volatilities

Bond prices depend on the covariance matrix $V$ only through the affine term $v$, and therefore

$$\frac{\partial\ln(P)}{\partial V} = {1\over 2}\int_0^T u\,u^\prime\,\mathrm{d}\tau\quad .$$ (33)

Given $n$ key-rates the model-covariance matrix $V$ can be mapped to the key-rate covariance matrix via $V_\mathrm{kr} = \mathsf U_{\mathrm{kr}}^-\,V\,{\mathsf U_{\mathrm{kr}}^-}^\prime$. This gives

$$\frac{\partial\ln(P)}{\partial V_\mathrm{kr}} = {1\over 2}{... ...u\,u^\prime\,\mathrm{d}\tau\,\mathsf U_{\mathrm{kr}}^-\quad .$$ (34)

With factor volatilities $\sigma_i:=\sqrt{V_{ii}}$ and diagonal volatility matrix $S:=\mathrm{diag}(\sigma)$ the correlation matrix reads $C=S^-\,V\,S^-$. This gives yield-sensitivities to volatility and correlation

$$\frac{\partial y(T)}{\partial\sigma_i} = -{1\over T}\int_0^T (C\,S\,u)_i\,u_i\,\mathrm{d}\tau\quad ,$$ (35)

$$\frac{\partial y(T)}{\partial C} = -{1\over 2}\,S^-\left[{1\over T}\int_0^T u\,u^\prime\,\mathrm{d}\tau\right]\,S^-\quad .$$ (36)

The sign of the yield-sensitivities depends crucially on the factor-correlations.