Bootstrapping OIS and Libor Curves from Basis Swaps
Summary
The discussion explains how overnight index swaps and Libor–OIS basis swaps relate to interest-rate curve construction. An OIS floating coupon compounds daily overnight fixings over the accrual period and annualizes the resulting compounded return. A basis swap exchanges two floating-rate references; its quoted spread can be understood through the difference between the corresponding swap rates under the relevant convention.
For modern multi-curve setups, the more detailed answer describes bootstrapping an OIS discount curve first, then deriving a standard-tenor projection curve using swaps, and building other tenor curves from basis or direct swaps. The OIS curve is used for discounting while the projection curves estimate forwards. Exact instruments and interpolation choices vary by currency, market and system, and different curve representations may match observed prices while producing different risk decompositions. The replies provide a framework, not a complete numerical bootstrap specification.
Key ideas
- An OIS floating coupon compounds daily overnight index fixings across its accrual period.
- A Libor–OIS basis swap exchanges floating-rate references and quotes a spread between them.
- In a multi-curve framework, bootstrap the OIS discount curve before projection curves.
- Projection curves for different tenors may be derived from standard swaps, basis swaps, or direct swaps.
- Curve instruments and implementation choices depend on market conventions and can change risk attribution.
Tags
Full text
# Libor OIS basis swap equation
# Libor OIS basis swap equation
I'm a little embarrassed about this because I have a PhD in math, but I'm having a little trouble working out how to bootstrap an OIS curve from libor rates and basis swap rates. If I had an equation for valuing everything I'd be set. What I have is a libor curve and a mysterious rate (US dollar OIS) coming from Bloomberg that is somehow related to basis swaps. I have been told that the basis swaps are between Libor and OIS, and I don't know what the equation for that looks like.
For some background, I just learned how to value a swap today, and from that I was able to figure out how to bootstrap the libor curve. I just need an equation to fit the numbers into. I've gotten loads of verbal explanations, so no need to go out of your way to provide one. Thanks.
## Answer by Physcs Envy (score 6, accepted)
https://quant.stackexchange.com/a/18993
An OIS, or Overnight Index Swap, is an interest rate swap whose floating leg payments are calculated as a geometric average of the daily fixings of some underlying O/N or T/N index (these indices are generally volume-weighted averages of reported daily transactions). The annualized floating leg rate is defined as $$ c_T^{float} = \frac{\prod^{s+T}_{t=s}{(1+r_t\delta (t))}-1}{\delta (T-s)}, $$ where $s$ is the first fixing day of the coupon period and $T$ is the last. $r$ is the value of the underlying index at time $t$, and $\delta (\cdot)$ is the year fraction according to an appropriate day count convention.
A basis swap is an exchange of one floating rate for another. In this case, it refers to the exchange of USD Libor for USD OIS or vice versa. Generally, if we refer to the two bases in a basis swap as $\alpha$ and $\beta$ and you are fixed payer in the swap with basis $\alpha$, your basis swap rate (or fixed rate spread) is $$ r_{basis swap} = c^{fixed}_\beta-c^{fixed}_\alpha, $$ where $c_b^{fixed}$ is the fixed coupon rate for a interest rate swap with basis $b$. From here on out, you may derive your implied OIS rates from the two rates and bootstrap as normal.
## Answer by achirikhin (score 1)
https://quant.stackexchange.com/a/79267
One should note that the exact implementation can be bank/system dependent, but the general idea in the OIS/Libor world was
- First bootstrap OIS curve. It is a self-discounting curve, i.e. both discount factors and forward are computed using same curve. Conceptually, it replaces the self-discounting Libor curve.
- Assuming perfect collateralisation in the same currency, strip standard projection curve (3m for US, 6m for EUR and GBP etc), using the standard IRSs for such fixed/floating swap. It was currency dependent. E.g. for USD it was 3m float vs 6m fixed, for Euro it was 6m float vs 1y fixed etc. Ois curve, stripped at step 1) will be used for discounting.
This gives you the "standard" tenor projection (forward) curve in corresponding currency, assuming cash collateral hence OIS discounting.
- Now you can strip projection curves for all other non-standard tenors, e.g. 1m or 6m or 12m for USD.
The products from which you will be stripping will strongly depend on the currency. It can be a basis floating/floating swap against the standard tenor (e.g. 3m/6m swaps for USD), or direct swaps against 6m USD. Ois curve, stripped at step 1 will be used for discounting in ALL cases, only projection curve will be built.
Importantly, the internal implementation of such projection curve can be either as a yield curve on its own (yield or DF interpolator) or a basis curve to another curve, e.g. standard tenor swap. The pricing result will be same for the observable swaps, but risk decomposition will be different. Practically, this is decided based on what instruments really drive the market, basis swaps (in which case non standard projection curve would be a basis curve) or straight swaps with non-standard basis. It is possible that different time segments of the curve are interpolated differently because of that.
To conclude, prod-grade set up of the discount curve is very different from what you see in the textbooks.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.