Why Derivatives Use Separate Forward and Discount Curves

Summary

The document explains why equity derivatives may use one curve to project forward rates or determine payoffs and another to discount cash flows. In a simplified theoretical argument, borrowing at an overnight rate and lending at LIBOR could appear to create an arbitrage spread within a dual-curve framework.

The answer says this opportunity is not practically available as a straightforward arbitrage. For collateralized derivatives, overnight indexed swap discounting reflects the funding value of collateral, which is generally tied to overnight rates. LIBOR can still serve as a benchmark for the derivative payoff. For non-collateralized trades, the appropriate funding treatment is less settled; the document notes debate around how mark-to-market funding should be handled. It offers a conceptual explanation rather than quantitative evidence or a procedure for measuring the spread in a specific market.

Key ideas

• Dual-curve pricing separates the curve used for payoff benchmarks from the discounting curve.
• A theoretical spread trade would borrow at overnight rates and lend at LIBOR.
• Collateralized derivatives are commonly discounted using OIS rates because collateral is overnight funded.
• The funding treatment for non-collateralized derivatives remains a debated issue.

Tags

# Arbitrage when discounting and forward computation is done with different curves


# Arbitrage when discounting and forward computation is done with different curves












I notice that (equity derivatives) trades generally are priced with different forward curve and discounting curve, which clearly lead to arbitrage. Is this arbitrage value too small to be ignored? How is it managed?

## Answer by siou0107 (score 3, accepted)

https://quant.stackexchange.com/a/50831

The last edition (10th, 2017) of Hull's book explains it fairly well. Basically, there is indeed a theoretical arbitrage within the dual curve framework: you could borrow at the overnight rate (Fed funds, SONIA, EONIA, etc.), lend at LIBOR and cash-in the spread in all your dynamic derivatives replication trades. However, such arbitrage is only theoretical : try to do that in practice!

As a matter of fact, the OIS discounting is relevant for collateralised derivatives since the derivative's value is funded by the collateral that is typically funded at the overnight rate. LIBOR is only useful as a benchmark to determine the derivative's payoff, and also (but there is still debate here) for non collateralised transactions since in such case, the derivative's mark-to-market is funded by the winning counterparty.

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.