Port a model dependent swaption sensitivity to a new model

Question

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 ?

Answer 1

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.