### What is the Swap Curve?

• What is the so-called Swap Curve, and how does it relate to the Zero Curve (or spot yield curve)?

Does it only refer to a curve of swap rates versus maturities found in the market? Or is it a swap equivalent of a spot-yield curve constructed from bootstrapping a bond yield curve?

The context of this question is set against a backdrop of a plethora of terminology (that seems to be used interchangeably). I am looking into how the so-called Zero Curve (or spot yield curve) is constructed in order to discount various IR derivatives (including swaps) when pricing them.

7 years ago

Garabedian,

Typically, the "swap curve" refers to an x-y chart of par swap rates plotted against their time to maturity. This is typically called the "par swap curve."

Your second question, "how it relates to the zero curve," is very complex in the post-crisis world.

I think it's helpful to start the discussion with a government bond yield curve to clarify some concepts and terminologies. Consider the US Treasury market, using the outstanding Treasury notes and bonds (nearly 300 of them...), we can either use bootstrapping or more sophisticated spline models to construct a "fitted curve." Since this yield curve represents bonds of identical credit risks (basically risk-free), the zero coupon curve, the discount curve, the forward curve, and the par yield curve are just different representations of the same thing and can be translated very easily from each other. For simplicity, I'll assume annual compounding:

• If you know the zero coupon rate $r_t$ for time $t$, then the discount factor is $1 / (1 + r_t)^t$.
• If you know the 1-year zero coupon rate $r_1$ and 2-year zero coupon rate $r_2$, then you can compute the 1-year forward 1-year rate from $(1 + r_1)(1+f_{1,1})=(1+r_2)^2$.
• You can also compute the 2-year par rate, just solve for $c$ from $$\frac{c}{(1 + r_1)} + \frac{100 + c}{(1+r_2)^2} = 100.$$

Now let's return to the swap market. To be concrete, let's consider a 2-year USD par swap. This instrument has four fixed leg payment, and eight floating payment. The par swap rate is the fixed-leg interest rate that sets the present value of all the cash flows to 0. In other words, we'd solve for the $c$ in: $$\sum_{i=1}^4 c \Delta_i d(T_i) = \sum_{j=1}^8 l_j \delta_j d(t_j),$$ where $d(t)$ is the discount factor for time $t$, $\Delta_i$ and $\delta_i$ are year fractions, and $l_j$'s are the 3M Libor forward rates.

Before the financial crisis, it is assumed that the discount curve and the forward curve are both based on Libor. This simplifies things a lot – just build a Libor forward curve so that it reproduces libors, futures rates, and par swap rates, and you're done. In this framework, all the translations (from zero curve to par curve to forward curve, etc.) above are still valid.

Unfortunately, the idea that Libor was the appropriate funding rate was completely invalidated during the crisis. In recent years, a common practice is to use the "OIS discounting"-based "multi-curve" approach. In the equation above, the $l_i$'s are still based on the 3M Libor forward curve, but the $d(t)$'s should be discount factors fitted to overnight indexed swaps.

Simply put, when you are building a swap curve, you now need to simultaneously calibrate both the OIS discount curve AND and Libor discount curve... Under this new paradigm, the simple translation that we used for government bonds above no longer works, since multiple curves are involved.

But it gets worse... since 1M Libor and 3M Libor have different credit risks, you can't even do something like $(1 + \text{Libor}_{\rm 1M}/12)(1 + \text{Libor}_\text{1 month forward 2 month} / 2) = 1 + \text{Libor}_{\rm 3M} / 4$! Instead, you need to build separate 1M and 3M Libor forward curves to account for the tenor basis...

As you can see, building a swap curve nowadays is a pretty involved task. What we now refer to as "a" swap curve is actually a collection of curves (OIS curve, 1m libor, 3m libor, 6m libor, etc.) bundled together...

There are numerous literature you can find on this topic just by googling "multi-curve". For example, http://developers.opengamma.com/quantitative-research/Multiple-Curve-Construction-OpenGamma.pdf