Repo Financing and Two-Curve Swap Pricing (Chapter 17)
Book: Fixed Income Securities: Tools for Today’s Markets | Author: Bruce Tuckman & Angel Serrat | ISBN: 978-0-470-89169-8
Previous: Interest Rate Swaps Valuation (Chapter 16) | Next: Caps, Floors, and Fixed Income Options (Chapter 18, Part 1)
Chapter 1 made discounting look clean. Chapter 17 reminds you that real traders borrow and lend through repo, post swap collateral at fed funds, and cannot always finance interim losses. The old math still works, but only under conditions that often fail in practice.
Bonds with Repo Change the Arbitrage Story
Shorting a bond through repo generates no cash today. You sell the bond, lend the proceeds as collateral, and borrow cash on the long side through another repo. Initial cash flow is zero, not the mispricing you thought you pocketed.
If repo rates match on both legs and prices converge, profit at unwind equals the future value of the initial mispricing at the repo rate. That is Scenario I.
Scenario II is brutal. The cheap bond finances at general collateral 0.149%. The rich replicating portfolio is special at 0%. You lose money even as relative mispricing goes to zero. Richness might just be expensive borrow.
Scenario III adds path risk. Mispricing widens. You must inject cash to roll repos or unwind at a loss. Hedge funds need capital buffers for this.
Discounting equals arbitrage only if all bonds finance at the same repo rate and you can finance divergences at that rate too.
Swaps: Collateral Rate Matters
Receiving fixed on a swap is like a financed bond, except the notional finances at LIBOR while the NPV finances at the collateral rate (usually fed funds, not LIBOR).
Replication of a 5% one-year swap off par swaps works when collateral earns the same rate as the floating index. If collateral earns 25 bp while LIBOR is 70 bp, the “replicating” portfolio fails. Scenario II in the swap section proves it with numbers.
So non-par swaps cannot be priced off par swaps with one curve unless collateral equals the floating index.
Two-Curve Pricing After the Crisis
Take fed funds / OIS as the riskless investable rate. OIS floating legs are par. Price any LIBOR swap by replication:
- Fixed vs. LIBOR swap you want to price
- Fixed vs. fed funds OIS swap
- LIBOR vs. fed funds basis swap at market spread
The LIBOR swap NPV equals the OIS swap NPV adjusted for the basis. Collateral earns fed funds on all three.
In practice you run two curves. Discount everything at OIS. Project LIBOR using basis-adjusted forwards built from par LIBOR swap rates or explicit basis spreads. Industry standard since 2010.
Errors from single-curve LIBOR discounting were small pre-crisis (a few bp on liquid swaps) but grew large when LIBOR-OIS blew out. Receivers of above-par fixed who hold collateral benefited from the bug.
Why This Chapter Hits Hard
It connects Chapter 1 repo (Chapter 12) to Chapter 16 swaps to Chapter 15 LIBOR-OIS in one thread. You cannot understand modern swap desks without two curves. You cannot understand why European sovereigns trade negative OIS spreads and positive LIBOR spreads without the bond spread table in Chapter 19 either, but the mechanics start here.
Tuckman includes formal proofs in the appendix. The prose version is enough for intuition: financing rates and collateral rates are not details. They are part of the price.