Interpolating VIX Risk-Free Rates Across Option Expirations
Summary
The document asks how to derive the two risk-free rates used in VIX calculations from a table of Treasury yields when the near-term and next-term SPX option series expire on different days. It describes linear interpolation between the 30-day and 60-day yields, followed by conversions from bond-equivalent yield to an annual effective rate and then to a continuously compounded rate using a logarithm.
The author gives example inputs and calculated values for both maturities, then notes that these results do not match expected rates. No answer or verification is included, so the source does not identify the error or establish whether the interpolation inputs, day-count convention, yield conversion, or target rates are appropriate. It is useful as a worked question about translating quoted Treasury yields into VIX methodology inputs, but readers must consult the cited methodology and independently check the calculations before applying them.
Key ideas
- The question concerns interpolating Treasury yields to match two option expiration maturities.
- It converts the interpolated bond-equivalent yields to annual effective rates and then to continuously compounded rates.
- The example calculations do not match the expected rates stated by the author.
- The document provides no resolution, so the correct procedure and source of the discrepancy remain unverified.
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Full text
# VIX methodology, risk-free interest rates R1 and R2 # VIX methodology, risk-free interest rates R1 and R2 I am trying to calculate the risk-free interest rates R1 and R2 for the data that is given in this pdf: https://cdn.cboe.com/api/global/us_indices/governance/VIX_Methodology.pdf using the methodology described in the secction 2a) of this pdf: https://cdn.cboe.com/api/global/us_indices/governance/Cboe_Volatility_Index_Mathematics_Methodology.pdf We have this fixed maturities in days and the respective rates: | Maturity days | yield rates | | 30 | 0.03% | | 60 | 0.02% | | 91 | 0.04% | | 182 | 0.05% | | 365 | 0.08% | | 730 | 0.11% | | 1095 | 0.22% | | 1825 | 0.59% | | 2555 | 1.00% | | 3650 | 1.37% | | 7300 | 2.03% | | 10950 | 2.21% | and that the near-term SPX constituent series expire in 24 calendar days and the next-term SPXW constituent series expire in 31 calendar days. For R1 I did: t = 24 CMT_i = 0,03% t_i = 30 CMT_i+1 = 0,02% t_i+1 = 60 Hence: Interpolated_CMT_for_24_days = CMT_i + (t - t_i)/(t_i+1 - t_i) * (CMT_i+1 - CMT_i) = 0,03% + (24 - 30)/(60 - 30) * (0,02% - 0,03%) = 0,032% = BEY_1 And using the formula descibred before Section 3 of the 2nd pdf, then APY_1 = (1 + BEY_1/2)**2 - 1 = 0,000320025599999774, and R1 = ln(1 + 0,000320025599999774) = 0,00031997 aproximated For R2 I did: t = 31 CMT_i = 0,03% t_i = 30 CMT_i+1 = 0,02% t_i+1 = 60 Interpolated_CMT_for_31_days = CMT_i + (t - t_i)/(t_i+1 - t_i) * (CMT_i+1 - CMT_i) = 0,03% + (31 - 30)/(60 - 30) * (0,02% - 0,03%) = 0.00029666666 = BEY_2 APY_2 = (1 + BEY_2/2)**2 - 1 = 0,000296688662776878, and R2 = ln(1 + 0,000296688662776878) = 0,000296645 aproximated. But I should have gotten R1 = 0.00031664 and R2 = 0.00028797. Can someone please identify where are the errors?
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