How OIS, Discount, Swap, and Basis Curves Relate in Option Pricing
Summary
The document distinguishes several USD interest-rate curves that may appear in equity option pricing. A discount curve supplies discount factors for cash flows, but its name alone does not specify the benchmark or collateral convention used to construct it. For cleared, collateralized derivatives, the text identifies the Fed Funds overnight indexed swap curve as the common discounting standard.
An OIS swap curve reflects swaps whose floating benchmark is the daily Fed Funds rate. “Basis swap” is less specific: it may refer to swaps between different floating benchmarks in one currency or to cross-currency swaps. The curves are related through a multi-curve framework: the Fed Funds discount curve can be derived from OIS swaps, while market practice may jointly solve for curves using liquid swaps and LIBOR/Fed Funds basis swaps. The answer is a terminology guide rather than a full pricing recipe; the applicable curve depends on the instrument’s cash flows, collateral, and model conventions.
Key ideas
- A discount curve provides discount factors, but the benchmark used to build it must also be specified.
- The Fed Funds OIS curve is a common discounting curve for cleared, collateralized derivatives.
- A basis swap can refer to either a single-currency benchmark spread or a cross-currency basis.
- Liquid swaps and basis swaps can be used together to construct a multi-curve framework.
Tags
Full text
# Interest rate curve in option pricing # Interest rate curve in option pricing 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? - US Swap Curve - USD Discount curve - USD SWAP OIS Fed Funds rate - USD Basis Swap ## Answer by Attack68 (score 1, accepted) https://quant.stackexchange.com/a/35969 - 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. - 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. - 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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