Chapter 34: Swaps Revisited

Variations on the Vanilla Deal

The plain vanilla interest rate swap (fixed-for-floating) can be modified in many ways:

Compounding Swaps

In a compounding swap, the floating rate compounds over the accrual period (instead of being paid each period separately). The floating payment at the end of a compounding period is based on:

$\text{Floating Payment}=L\cdot \left[{\prod }_{i=1}^n(1+R_i{\tau }_i)-1\right]$

where $R_i$ are the sub-period floating rates and ${\tau }_i$ are the sub-period lengths. This reduces payment frequency while preserving the economic equivalence (approximately).

Valuation requires a convexity adjustment for compounding swaps, similar to the adjustment for LIBOR-in-arrears.

Currency and Nonstandard Swaps

Differential Swap (Diff Swap / Quanto Swap)

The floating leg is based on an interest rate in one currency but applied to a principal in another currency. Example: Receive SOFR applied to a JPY principal.

Pricing requires adjustments for:

• The correlation between the interest rate and the exchange rate
• The quanto effect (the payoff is in a different currency than the rate's natural habitat)

Cross-Currency Basis Swaps

Exchange of floating rates in different currencies plus the exchange of principal. The spread between the floating rates is the cross-currency basis. Historically, this basis was close to zero (covered interest parity). Post-2008, significant and persistent basis exists, reflecting:

• Different demand for funding in different currencies
• Regulatory constraints on bank balance sheets
• Costs of hedging currency risk

Equity Swaps

An agreement to exchange the total return on an equity index (or individual stock) for a fixed or floating interest rate.

$\text{Equity leg payment}=N\cdot \left(\frac{S_{t_i}-S_{t_{i-1}}}{S_{t_{i-1}}}\right)$

If the return is negative, the equity receiver pays the equity payer.

Applications:

• Foreign investors gaining exposure to a market without owning the stocks directly
• Hedge funds implementing market-neutral strategies
• Index fund managers adjusting exposure without trading the underlying basket

Swaps with Embedded Options

Cancelable (Callable) Swaps

One party has the right to cancel the swap at predetermined dates. Equivalent to a plain vanilla swap + a swaption. Valuation: price the plain vanilla swap and the embedded swaption(s) separately.

Extendable Swaps

One party has the right to extend the swap beyond its stated maturity. Equivalent to a plain vanilla swap + a forward-starting swaption.

Index Amortizing Swaps

Principal amortizes based on the level of a reference rate. When rates are low, principal amortizes faster (mimicking mortgage prepayment).

Swaptions

A swaption gives the right to enter into a swap at a future date. Already covered in Chapter 29. Payer swaption = option to pay fixed. Receiver swaption = option to receive fixed.

Other Swap Structures

Key Valuation Principle

All swaps can be valued as:

1. Two streams of cash flows discounted at the appropriate risk-free rate
2. With appropriate adjustments for credit risk (CVA/DVA), funding costs (FVA), and other XVAs
3. For exotic features (callability, extendability), separate into the vanilla swap and the embedded option(s)