# 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.

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."
• Thanks for the information. If I would want to calculate these forward rates from zero-coupon bond prices, where should I look for data? – user7778 May 12 '14 at 16:36