Interpolating Cross-Currency Basis Curves Through Discount Factors
Summary
The document describes a market practice for estimating cross-currency basis between quoted maturity dates. Rather than interpolating the quoted basis or outright FX forwards directly, it presents a representation of the FX forward using spot, domestic and foreign OIS discount factors, and a cross-currency basis discount factor. The basis discount curve is bootstrapped from FX swaps and cross-currency swaps, then interpolated to obtain values at unquoted maturities.
For interpolation, the discussion cites approaches used for other curves, such as linear interpolation on yields or on log discount factors. It cautions against splines because their nonlocal behavior can create sensitivities to distant instruments; the example is a shorter-dated trade reacting to a much longer-dated quote. This is an industry-practice account, not a universal convention: curve construction and interpolation choices may depend on the system and market, and no comparative empirical results are provided.
Key ideas
- Bootstrap the cross-currency basis discount curve from FX swaps and cross-currency swaps.
- Use the basis discount factor together with OIS discount factors and spot to represent FX forwards.
- Common interpolation choices include linear interpolation on yields or log discount factors.
- Nonlocal spline interpolation can create sensitivities to distant curve instruments.
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Full text
# Interpolating cross-currency basis curve
# Interpolating cross-currency basis curve
Just wondering how do people "interpolate" between different "pillar dates" on a cross-currency basis curve? So say for example, if the observed spot is 1.5, observed CC basis for 9 months is -1.25 and CC basis for 1 year is -1.35, and I am trying to work out, say, the cross-currency basis for, say, 10 months, how should I do that?
A few ideas I can think of:
- Interpolate between -1.25 and -1.35 (which ended up say, around -1.30) and call that my CC basis.
- Interpolate between the outright rates derived from the basis and spot (i.e., in the example above, between 1.5-1.25/100 = 1.4875 and 1.5-1.35/100 = 1.4865) and then subtract the interpolated value (which is the forward rate) with the spot?
- Interpolate on the IR curves of r_f and r_d and use parity to derive the forward rates and subtract that with spot?
Or maybe something else?
What is the "correct" way of doing it and how does people do that in general in the industry?
Finally, for the "correct" way, what type of interpolator is the conventional one to be used? Linear? Cubic spline?
## Answer by Antoine Conze (score 0, accepted)
https://quant.stackexchange.com/a/38877
The most common way I have seen in front office systems is to interpolate/bootstrap the CC basis zero curve $D^{CC}(T)$, defined from the following representation $$ FX^{fd}(T)=FX^{fd}(0) \frac{D^f_{OIS}(T)}{D^d_{OIS}(T)} D^{CC}(T) $$ where $FX^{fd}(T)$ is the forward FX, $D^d_{OIS}(T)$ is the domestic OIS discount factor and $D^f_{OIS}(T)$ is the foreign OIS discount factor.
$D^{CC}(T)$ is bootstrapped on market FX swaps and XCCY. Interpolation of $D^{CC}(T)$ tends to be along the same lines than that of other curves, i.e. linear on yields or linear on log discount. Splines are always a problem as they do not form a local interpolation method, and thus generates nonsensical sensitivities (e.g. a 5 years deal having non zero sensitivity to a 10 year instrument).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.