# Port a model dependent swaption sensitivity to a new model

I have a short rate model in which I have (among others) the following metric (for leverages) for a swaption : $$L = \frac{\frac{\partial}{\partial r}V_0^{\textrm{Swaption}}}{\frac{\partial}{\partial r}V_0^{\textrm{Swap}}}$$ where $$V_0^{\textrm{Swaption}}$$ is the time $$0$$ swaption model price and $$V_0^{\textrm{Swap}}$$ the time $$0$$ model price of the underlying swap.

Now I am moving from this short rate model to a model where a dynamic is prescribed on the forward swap rate, imagine for instance a lognormal Black model on the forward swap rate, to make it simple. I would like to port the metric $$L$$ to the new model.

The quantity $$\frac{\partial}{\partial r}V_0^{\textrm{Swap}}$$ is, I guess, nothing but the annuity (a.k.a. level or pvpb) of the underlying swap, equal to $$\sum_i \delta_i P_{0,T_i}^d$$ and given by the spot discount curve (and interpolation).

But how to port the quantity $$\frac{\partial}{\partial r}V_0^{\textrm{Swaption}}$$ to the new model though ? Of course, I can write $$\frac{\partial}{\partial r}V_0^{\textrm{Swaption}} = \frac{\partial s_0}{\partial r} \frac{\partial}{\partial s}V_0^{\textrm{Swaption}}$$ where $$\frac{\partial}{\partial s}V_0^{\textrm{Swaption}}$$ is the delta in the new model (w.r.t. the swap rate $$s$$), but what to do with $$\frac{\partial s_0}{\partial r}$$ quantity ?

If the port is not possible, what would be an equivalent metric in the new model ?

We note $$A$$ the annuity, so that $$V^{swap} = A(s - K)$$ so that $$\frac{\partial V^{swap}}{\partial s} = A$$. As the chain rule gives $$\frac{\partial V^{swap}}{\partial r} = \frac{\partial s}{\partial r} \frac{\partial V^{swap}}{\partial s}$$ we get that $$\frac{\partial s}{\partial r} = \frac{1}{A} \frac{\partial V^{swap}}{\partial r}$$ and as the chain rule gives also : $$\frac{\partial V^{swaption}}{\partial r} = \frac{\partial s}{\partial r} \frac{\partial V^{swaption}}{\partial s}$$ which combined with the previous equation gives $$\frac{\partial V^{swaption}}{\partial r} = \frac{1}{A} \frac{\partial V^{swap}}{\partial r} \frac{\partial V^{swaption}}{\partial s}$$ and diving both sides by $$\frac{\partial V^{swap}}{\partial r}$$ gives $$L = \frac{1}{A} \frac{\partial V^{swaption}}{\partial s}$$ where $$A$$ is given by the spot discount curve and $$\frac{\partial V^{swaption}}{\partial s}$$ by the new model.