Pricing and risk management of basis swaps is more nuanced than plain-vanillas. For one reason, the primary risk captured by them is the credit risk between the indices, say Government (Fed-Funds) versus Bank (Libor) credit. They also at times represent supply-demand dynamics of one index versus another one, independent of the inherent credit risk in each index. In order to provide a consistent framework to price them, one has to step back from the previous setup, where a single discount factor curve captured both discounting and calculation of forward rates, as basis swaps provide independent information about the forward/locking curve of an index, separate from the discount (funding) curve.

Notice that there are two dates here: trading date t, and forward delivery date T > t. On any given trading date t, we can graph F_A(t, ·) as a function of forward dates T, and come up with forward curve of the asset at t. Obviously, if T = t, then we have the spot/cash price of the asset: F_A(t, t) = A(t). We can think of the forward curve as the indifference curve of the asset: Given all the information today, we are indifferent between paying A(t) for the asset today, or agreeing to pay F_A(t, T) at some future delivery date T.

For a fixed forward delivery date T, the forward price at any trading day t ≤ T will fluctuate depending on market conditions. At the delivery date, of course, the forward price will coincide with the spot price: A(T) = F (T, T). Since the forward price fluctuates at any trading day, the value of a given (seasoned) forward contract with a fixed delivery price K will also fluctuate. For example, assume it is January 1, and an asset is trading at 100, and its forward value for delivery on March 31 is 101. We might enter into this contract at 101 with no exchange of money (just a handshake). For the next 3 months, the value of this contract will fluctuate, depending on each day′s market perception of the March-31 (forward) delivery price of the asset. At the maturity of the contract, if the spot price of the asset is higher than 101, then the contract has positive value, since it enables a long to gain the asset for lower than its market value. Similarly, if the spot price is lower than 101, then the contract has negative value, since it obligates the long to buy an asset for higher than its market value. Depending on the contract, one can either cash-settle, or take/make physical delivery of the asset at the agreed rather than the actual spot price.

One might think that determining the T-forward value would involve forecasting of the T-realized price of the asset. However, a simple cash-and-carry argument shows that we can determine the forward value of an asset without resorting to forecasting at the inception of the contract, t, he has to deliver the asset at time T in exchange for the (to be determined) forward value K = F_A(t, T). He can conceptually buy that asset today t at A(t) by taking a loan—potentially collateralized by the asset—with maturity T and hold on to the asset till maturity. At maturity, he will deliver the asset, receive the K (agreed upon at time t and fixed thereafter), and repay the A(t) loan plus the interest. As long as K equals the loan and interest, he will have no risk. So the forward price, K, must equal loan plus its interest. If the asset can only be carried via an uncollateralized risk-free loan, then the forward prices can be related to risk-free interest rates or, equivalently, their implied discount factors:

In general, the cash-and-carry argument shows that the forward value of an asset is the spot value plus the cost of carrying the asset minus any income that accrues to the holder of the asset, properly future valued to the forward date. For example, the forward gross value of a bond is its spot gross value plus the repo (carry cost) minus the future value of any intermediate coupon income. For dividend-paying stocks with known discrete dividends we need to subtract the future-value of the dividends from the carrying cost.

The second way that forward rates arise is as a contract based on only the interest component of a hypothetical loan, with the future interest rate locked at some level K. For example, a USD FRA—equivalent to a single-period forward swap—is simply the interest component of a nominal loan at the future setting of the specified index, r_ΔT(T) (read Libor-3m), versus that of a locked (fixed) level K: paid at T + ΔT, with K chosen so that this contract has zero value today. We previously argued that as long as we can invest future moneys at the same index, that is, our funding is based on r_ΔT, we can replicate the unknown future cash flow by owning/owing unit currency at start/end of the loan. Hence the value of floating payment paid at T + ΔT equals which means that the rate K of a zero-cost FRA is the same as the implied forward rate. Therefore, if we already have our funding/discount curve, we can use it to calculate forward locking rates for the index. Alternatively, and more commonly, given a series of forward locking rates for our funding index, we can extract our funding/discount curve. For example, if we fund ourselves at some index X periodically, our funding/discount curve becomes: where K _X([T_i , T_{i +1} ]) is the locking (FRA) rate for the forward period [T_i, T_i+1] with 0 = T₀ < T₁ < . . . , and ΔT_n = T_n+1—T_n.

Still, one can focus on the expected difference between the index X(T) and the lockable funding index r (T). The payment of an FRA based on an index X is X(T) - K_X properly accrued for the duration of the hypothetical loan, ΔT, and paid at T + ΔT. This can be reexpressed as: where K_X = K_X-r + K_r . To determine the K_X that will make the value of this rate-lock agreement zero, since we fund ourselves at r , we know how to calculate K_r so that the value of the second term becomes zero. Therefore, we just need to focus on selecting K_X—r, that is, the locking level for our basis risk: the exposure to the difference between X and our funding index, r.

Basis swaps are locking levels for forward basis risks, and their market quotes are based on expected future difference (spread) between the indexes. These market consensus levels are primarily based on market′s forecast of future credit spreads—usually projected forward from their historical levels—modulated by supply-demand dynamics. For example, the difference between Fed-Funds rate versus Libor is primarily U.S. Government credit versus banking credit. In times of good financial health for banking, this spread runs around 15 bp, and as such will be the expected spread as captured in Fed-Fund-Libor basis swaps. In times of turmoil, however, it can easily blow out.

Fed-Funds versus 3M-Libor: Weighted arithmetic average of overnight Fed-Funds effective rate plus spread versus 3m-Libor, both legs paid Q, Act/360.

Prime versus 3M-Libor: Weighted average of daily resets of prime rates taken from the Fed statistical release H.15 minus spread versus 3m-Libor, both legs paid Q, Act/360.

Commercial-Paper (CP) versus 3M-Libor: Unweighted monthly average of daily resets of 1-month CP rate taken from the Fed statistical release H.15, converted from discount basis to Act/360, compounded monthly at CP flat, minus spread versus 3m-Libor, both legs paid Q, Act/360.

T-Bills versus 3M-Libor: Average of the bond-equivalent yield of the weekly auction average of the 3-month U.S. T-Bills taken from the Fed statistical release H.15 plus spread versus 3m-Libor, both legs paid Q, Act/360.

Basis swap based on constant-maturity-treasury (CMT) indexes published in H.15 used to be quoted, but these are now backed out from CMS swap level and spread-locks, although care has to be taken to adjust for the difference between a CMT index versus the U.S. current (CT) yields.

Table 4.1 shows sample quote sheets for a few of these swaps.

A consistent framework for capturing, pricing, and risk management of plain-vanilla and basis swaps emerges as follows:

It might seem strange that these swaps exist, since by prior arbitrage arguments, the compounded shorter-term rate must equal the longer rate, and the arbitrage-free spread should be zero. For example, the periodic payoff of a 3′s/6′s with a spread of s for each calculation period [T, T + 6m] is the net payment at T + 6m of versus or equivalently

Since there is no net exchange of cash until maturity, the above relationship—being just a mathematical identity—must still hold even if all transactions were with a single risky bank. However, if we are borrowing, lending, and buying FRAs from different risky banks, it could happen that we lend money to a bank that subsequently defaults, while our other counterparties remain solvent, and we have to pay them. If we are dealing with counterparties of the same credit worthiness, the riskiest transaction is the longest loan as it has the longest default exposure window. Therefore, whoever is going to lend for 6m will quote/require a rate higher than what is implied by shorter-term (3m-rate, 3x6 FRA) rates.

To properly handle these basis swaps, we have to decide what our funding index is, say Libor-3m, and treat the other Libors (1m, 6m) just like other basis swaps, that is, extract a rate-lock curve for them so that the Libor-Libor basis swaps at market spreads price back to 0.

An overnight indexed swap (OIS) is a fixed-for-floating swap where the floating leg is based on an overnight index. The floating payment is typically the daily compounded interest at each overnight rate (weekends/holidays use previous business day′s fixing) over the calculation period, where α_i is the length of each compounding day according to some daycount basis (for example, α_i = 1/360 or 3/360 for Fridays in USD), while the fixed payment is the fixed rate accrued for the same calculation period, resulting in a single net cash flow for each calculation period. OIS swaps of maturity less than 1-year have one calculation period, while longer-term swaps are broken into annual calculation periods. Below is a list of OIS swaps and their benchmark index for different currencies:

USD: FFER (Fed-Funds effective rate) as calculated by the New York Fed, Act/360.

EUR: EONIA (Euro overnight index average), the effective overnight rate computed as a weighted average of all overnight unsecured lending transactions in the interbank market, calculated by European Central Bank (ECB) and published by the European Bank Federation, Act/360.

GBP: SONIA (Sterling overnight index average), the weighted average rate of all unsecured overnight cash transactions brokered in London by the Wholesale Markets Brokers′ Association (WMBA). SONIA closely follows the Bank of England′s (BOE) policy rate, Act/365.

JPY: TONAR (Tokyo overnight average rate), based on uncollateralized overnight average call rates for lending among financial institutions, published by the Bank of Japan (BOJ). BOJ affects TONAR using open market operations to keep it in line with its policy rate (called Mutan), Act/365.

CAD: CORRA (Canadian overnight repo rate average) released by the Bank of Canada, Act/365.

While this idea is gaining traction and acceptance among market participants, the market practice is to still use Libor rather than OIS discounting when creating projection curves out of quoted market swaps. At the same time—and inconsistently—when offering unwind prices for existing/ seasoned swaps with large mark-to-market value, the actual funding curve is conveniently remembered/invoked, usually (and not surprisingly) by the benefiting party!

As OIS swaps gain more liquidity and are traded for longer terms—there are currently liquid points for at most a few years—a more consistent framework for building Libor curves and pricing swaps should emerge. Until then, most at-market swaps would be priced and discounted off of the Libor curve, while off-market swaps with large mark-to-market value would use OIS or similar discount curves.