Using Zero-Coupon Curves to Obtain Historical Instantaneous Forward Rates
Summary
The question concerns historical rate data needed to estimate volatility in a model using instantaneous forward rates at different observation dates and relative maturities. The response says the referenced chart already reports instantaneous forward rates in the required date-and-maturity form, rather than only spot-date zero rates. It also describes how such curves are obtained: zero-coupon yields are estimated from traded government bond prices and yields, since hypothetical zero-coupon instruments are not directly observable across the full maturity range.
The cited European curve construction uses euro-denominated central government bonds and zero-coupon bonds, while excluding instruments with variable coupons or special features. It screens for minimum issue size, trading activity, bid-ask spreads, and maturities, and removes yield outliers within maturity brackets. These details clarify that a historical curve is a model-based estimate from selected securities, not a complete record of directly traded zero-coupon bonds. The answer does not provide a data download procedure or discuss how curve estimation choices affect a volatility study.
Key ideas
- Instantaneous forward-rate observations are indexed by observation date and future relative maturity.
- Zero-coupon curves are estimated from bond prices or yields because zero-coupon securities are not available across all maturities.
- The described ECB methodology filters eligible government bonds by instrument features, size, liquidity, maturity, and yield outliers.
- Curve data are estimated market measures, so sample selection and construction can affect downstream volatility estimates.
Tags
Full text
# Where can I find historical data for volatility estimation?
# Where can I find historical data for volatility estimation?
I'm trying to estimate volatility following Shreve book, so I need observations of $f(t_j,t_j+\tau_k)$ and $f(t_j+\delta,t_j+\tau_k)$, where $t_J<t_{J-1}<\dots<0$ and $\tau_k$ are relative maturities and f is inst. forward rate($\delta$ is small that $t_j+\delta<t_{j+1}$).
I found this one: http://tinyurl.com/83we3db It contains the inst. forward rate values, but are those data in the form of $f(t_j,t_j+\tau_k)$ or just $f(0,\tau_k)$ for every day.
Where can I find data for zero-coupon bonds to calculate these values? Thanks for help in advance.
## Answer by Matt Wolf (score 1, accepted)
https://quant.stackexchange.com/a/11256
The chart you linked to offers data for the "instantaneous forward rate" which are the rates you are looking for (f(tj,tj+τk)).
Regarding the construction of the zero-coupon yield curves (cited from the ECB website):
"The ECB estimates zero-coupon yield curves for the euro area and derives forward and par yield curves. A zero coupon bond is a bond that pays no coupon and is sold at a discount from its face value. The zero coupon curve represents the yield to maturity of hypothetical zero coupon bonds, since they are not directly observable in the market for a wide range of maturities. They must therefore be estimated from existing zero coupon bonds and fixed coupon bond prices or yields. The forward curve shows the short-term (instantaneous) interest rate for future periods implied in the yield curve. The par yield reflects hypothetical yields, namely the interest rates the bonds would have yielded had they been priced at par (i.e. at 100)."
and
"Selection of bonds The following criteria are applied when selecting bonds:
- Only bonds issued in euro by euro area central government (European System of Accounts 1995: sector code 'S.1311') are selected.
- Only bonds with an outstanding amount of at least € 5 billion are included. *Bonds with special features, including ones with specific institutional arrangements, are excluded.
- Only fixed coupon bonds with a finite maturity and zero coupon bonds are selected, including STRIPS . Variable coupon bonds, including inflation-linked bonds, and perpetual bonds, are not included.
- Only actively traded central government bonds with a maximum bid-ask spread per quote of three basis points are selected. The prices/yields are those at close of market on the reference day.
- In order to reflect a sufficient market depth, the residual maturity brackets have been fixed as ranging from three months up to and including 30 years of residual maturity.
- An outlier removal mechanism is applied to bonds that have passed the above selection criteria. Bonds are removed if their yields deviate by more than twice the standard deviation from the average yield in the same maturity bracket. Afterwards, the same procedure is repeated."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.