When pricing an equity option we calculate risk-free rate by interpolating one of the curves below for time-to-maturity T. What is the difference between the following curves and in what case each is applicable to option pricing?

  1. US Swap Curve
  2. USD Discount curve
  3. USD SWAP OIS Fed Funds rate
  4. USD Basis Swap
  1. USD Swap Curve: This is an ambiguous name, it could, in today's multi curve framework, mean multiple things. The most common layman definition of US Swap Curve is to represent the benchmark swap rate for different tenors on an x-y plane. E.g. 1Y: 1%, 2Y: 1.5%, 3Y: 2% etc. The benchmark is usually 6M LIBOR swaps in USD.

  2. USD Discount Curve. Again, an ambiguous name, but slightly less. A discount curve's use is to provide discount factors for valuing cashflows. Therefore that much is specified. On which measure to discount is a different matter (e.g. 3M LIBOR or 6M Libor or FFOIS or some arbitrary benchmark). The industry interbank standard for discount factors of cleared derivatives is the FFOIS discount curve.

  3. USD Swap OIS Fed Funds rate. The inclusion of the word swap means I should consistently equate this to 1. but instead of LIBOR swaps it is OIS Swaps whose floating rate benchmark is the daily FFOIS rate.

  4. USD Basis Swap. Again unclear since there is no context, but this could either represent a single-currency basis swap (SBS) (USD 3M LIBOR versus 6M LIBOR or 3M LIBOR versus FFOIS), or if referenced from a separate currency, e.g. EUR, it might reference the cross-currency basis swap (XBS), e.g. EUR/USD 10Y XCS.

The relation between these curves: For option pricing you are often interested in the forward valuation of cashflows, for which collateralised derivatives will be dependent upon the US FFOIS discount curve, which itself is derived from FFOIS swaps. More commonly the FFOIS rates are derived from the more liquid US Swaps and LIBOR/FFOIS Basis swaps which can all be simultaneously solved to produce a multi-curve framework.


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