Projecting Swap Risk onto Bonds and Futures

# Projecting swap risk on bonds and futures


# Projecting swap risk on bonds and futures












Let’s say I have a swap $S$. I understand the sensitivity of $S$ to the 5-year rate, where the 5-year rate is the compounded SOFR from today to 5 years in the future. This is simply the derivative $\frac{\partial S}{\partial r_5}$, as the swap PV can be expressed as a function $f$ of the 5-year rate: $S = f(r_5, \dots)$.

However, in practice on trading desks, swaps are often hedged using ONTR bonds and SOFR futures. Since swaps don’t have an analytical representation involving bonds (due to different discount curves), we cannot directly express $S$ as a function of bonds, i.e., $S \neq f(\text{BOND}_5, \dots)$.

My question is: how do you project the risk of a swap portfolio onto bonds when swaps cannot be written as a direct function of bonds?

More generally, if I have a portfolio of instruments $P_1$, how can I project its risk onto another set of instruments $P_2$, where elements of $P_1$ are not expressible as functions of elements in $P_2$?

One approach I’ve considered is using linear regression, with $P_1$ instruments as the target and $P_2$ instruments as the covariates. However, due to multicollinearity among the covariates, it becomes challenging to allocate risk efficiently. Are there more effective techniques to handle this?

## Answer by Attack68 (score 1)

https://quant.stackexchange.com/a/80903

> More generally, if I have a portfolio of instruments P1 how can I project its risk onto another set of instruments P2 where elements of P1 are not expressible as functions of elements in P2?

You need a mapping (https://en.wikipedia.org/wiki/Change_of_basis). Generally speaking this is a matrix transformation so that:

$$ \mathbf{S_2} = \mathbf{A S_1}$$

since $S_1$ and $S_2$ are risks, in the sense they are partial derivatives, then you have that:

$$ \frac{\partial V}{\partial \mathbf{r_2}} = \frac{\partial \mathbf{r_1}}{\partial \mathbf{r_2}} \frac{\partial V}{\partial \mathbf{r_1}}$$

When instruments can be directly related within a pricing model, e.g. forward swaps and par swaps, it is likely that $\frac{\partial \mathbf{r_1}}{\partial \mathbf{r_2}} $ is directly calculable within the same framework.

When you are working with different instruments the model is entirely subjective. E.g. suppose you have instruments $P_1$ which are gold and silver and you have instruments $P_2$ which are platinum and palladium. How do you create $\frac{\partial \mathbf{r_1}}{\partial \mathbf{r_2}} $? You need to subjectively design the derivatives of the prices of gold and silver with platinum and palladium respectively.

Using (multi-)linear regression is probably the place to start. If the overall objective is PnL attribution and minimising unexplained PnL, then any form of machine learning approach to this will probably devolve in its most basic form to some least squares regression anyway.