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Estimating Rates and Forwards for SPX Option Volatility Curves

Article Quant Q&A · Author: Tian
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Summary

The document presents two approaches to supplying rate and dividend inputs when constructing an implied volatility curve for SPX options. One approach estimates interest rates from put-call parity, incorporating the present value of known dividends for European options on dividend-paying stocks. This can provide an estimate for a given maturity from observed option prices and the underlying price.

A second approach follows the Cboe VIX methodology: interpolate and extrapolate Treasury constant-maturity yields with a cubic spline, convert them to continuously compounded rates, and use forward prices to avoid estimating a dividend rate directly. The forward is derived from call-put parity. These are methodological suggestions, not a comparison of accuracy. The document gives no data source details beyond the Treasury curve reference, nor implementation checks or treatment of noisy quotes. Its formulas and assumptions should be matched to the option style and market inputs being used.

Key ideas

  • Put-call parity can be rearranged to estimate an interest rate for a given option maturity.
  • Known dividends enter the parity relation for European options on dividend-paying stocks.
  • A VIX-style process interpolates Treasury yields and converts them to continuously compounded rates.
  • Using a parity-implied forward price can avoid estimating a dividend yield directly.
  • The document does not compare the accuracy of the two approaches.

Tags

OptionsEquitiesVolatilityDerivatives pricingFixed income
Full text
# interest rate, dividend rate data for black scholes model


# interest rate, dividend rate data for black scholes model












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!

## Answer by Bob Jansen (score 4)

https://quant.stackexchange.com/a/75399

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.

## Answer by nbbo2 (score 4)

https://quant.stackexchange.com/a/75400

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.

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.