I am working on a project to build an implied volatility curve for SPX options. But I am stuck with finding interest rate and dividend rate data for all maturities. Any recommended resources? Thanks!
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1$\begingroup$ Generally, if you get the market data for dividends you already have access to implied vol curves as well from that source (e.g. Bloomberg). $\endgroup$– AKdemyMay 1 at 7:11
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2 Answers
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The interest rate can be derived from put call parity. A number of questions about how to do this have been asked, for example this one but look at related questions as well.
For European options written on stocks with known dividends that will be paid out during the life of the option, the formula becomes (Wikipedia):
$$C(t) - P(t) + D(t) = S(t) - K \cdot e^{-r(T-t)}$$
Isolate $r$ to get an estimate of the interest rate for a given maturity.
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1$\begingroup$ Though in reality the by far hardest part is the dividend. Interest rates come from market quoted swap curves. $\endgroup$– AKdemyMay 1 at 6:03
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The experts on this issue are the people at the CBOE who compute the VIX volatility index. I suggest you use the same methodology described in this document Cboe Volatility Index Mathematics Methodology:
(1) For interest rates "The risk-free interest rate, 𝑟𝑡 , is calculated based on U.S. Treasury yield curve rates. The calculation process captures constant maturity Treasury (CMT) yields (i.e., bond equivalent yields) available on the U.S. Treasury website. Next a cubic spline is applied to interpolate/extrapolate a yield for each date between maturities, converts the bond equivalent yields (BEY) to annualized percentage yields (APY), and then converts these yields to continuously compounded interest rates for use in the Cboe volatility index calculation engine."
(2) Dividend rate: the need for a dividend rate (the hardest part) is bypassed entirely by using the Forward price $F$ instead of the stock price $S$ in the calculation of option prices (i.e. use the Black 76 formula instead of the Black Scholes formula for options). The Forward is found from the Put Call Parity relation as follows
𝐹 = Strike Price + 𝑒^𝑅𝑇 × (Call Price − Put Price)
More details are found in the above document.
