Rate Swaps

The implications of this new model class are in contrast to most term structure models discussed in the literature, which assume that the bond markets are complete and fixed income derivatives are redundant securities. Collin-Dufresne and Goldstein [ 18] and Heiddari and Wu [36] show in an empirical work, using data of swap rates and caps/floors that there is evidence for one additional state variable that drives the volatility of the forward rates. Following from that empirical findings, they conclude that the bond market do not span all risks driving the term structure. This framework is rather similar to the affine models of equity derivatives, where the volatility of the underlying asset price dynamics is driven by a subordinated stochastic volatility process (see e.g. Heston [38], Stein and Stein [71] and Schobel and Zhu [69]). 

Jagannathan R, Kaplin A, Sun S (2003) An Evaluation of Multi-Eactor CIR Models using LIBOR, Swap Rates, and Cap and Swaption Prices. Journal of Econometrics 116 113-146. 

There is still a consistency problem if we want to price interest rate derivatives on zero bonds, like caplets or floorlets, and on swaps, like swaptions, at the same time within one model. The popular market models concentrate either on the valuation of caps and floors or on swaptions, respectively. Musiela and Rutkowski (2005) put it this way We conclude that lognormal market models of forward LIBORs and forward swap rates are inherently inconsistent with each other. A challenging practical question of the choice of a benchmark model for simultaneous pricing and hedging of LIBOR and swap derivatives thus arises. ... 

Bond Price Swap Price Swap Rate( Redeu-uti ji (i)... 

In selecting the model, a practitioner will select the market variables that are incorporated in the model these can be directly observed such as zero-coupon rates or forward rates, or swap rates, or they can be indeterminate such as the mean of the short rate. The practitioner will then decide the dynamics of these market or state variables, so, for example, the short rate may be assumed to be mean reverting. Finally, the model must be calibrated to market prices so, the model parameter values input must be those that produce market prices as accurately as possible. There are a number of ways that parameters can be estimated the most common techniques of calibrating time series data such as interest rate data are general method of moments and the maximum likelihood method. For information on these estimation methods, refer to the bibliography. 

The traditional approach to yield curve fitting involves the calculation of a set of discount factors from market interest rates. From this, a spot yield curve can be estimated. The market data can be money market interest rates, futures and swap rates and bond yields. In general, though this approach tends to produce ragged spot rates and a forward rate curve with pronounced jagged knot points, due to the scarcity of data along the maturity structure. A refinement of this technique is to use polynomial approximation to the yield curve. 

This, therefore, dictates that a dealer bank should use the local currency OIS curve as inputs to construct a curve of projected Libor rates from current interbank swap rates. Swaps traded in euro currency would be priced off the EONIA curve, sterling swaps off the SONIA curve, and so on. 

The volatility value used can be estimated in two ways. We can estimate volatility separately and then use this to calculate what the approximate convexity adjustment should be, or we may observe the convexity bias directly and derive a volatility value from this. This would require an examination of market swap rates and bond yields, and use these to estimate the volatility implied by these rates. 

I-spread It is the yield spread of a corporate bmid relative to an interpolated swap rate. According to this approach, the yield of a corporate bond is built in the following way ... 

Craisider a hypothetical situation. Assume that an option-free bond paying a semi-annual coupon 5.5% on par value, with a maturity of 5 years and discount rate of 8.04% (EUR 5-year swap rate of 1.04% plus credit spread of 700 basis points). Therefore, the valuation of a conventional bond is performed as follows (Figure 9.4). 

German bonds in this case) rather than to any similarity between the swap rate and that of the peripheral country. 

Like the Eurodollar market, the Eurosterling sector is quite diversified by issuer type (Exhibit 6.16). Issuance of top quality paper with long maturities has been especially large, mainly because issuers can achieve very tight funding in LIBOR terms, given the relatively wide 30-year sterling swap rate. 

Strictly speaking, the FIAT 1 transaction does not generate excess spread. This explains the high level of credit enhancement from the unrated class M notes (usually, unrated tranches are either privately sold or kept as an equity tranche by the originator). On the closing date, an amount of notes was issued which was equal to the net present value of all future cash payments due from the collateral (as opposed to the principal balance of the collateral). The discount rate used was the fixed rate payable to the swap counterparty (swap rate plus coupon on the class A notes and all fees associated with the transaction). Structured this way, the receivables always yield the discount rate, leaving no excess spread in the transaction. However, losses on the FIAT 1 portfolio can be covered to a certain degree from interest collections because the structure provides for delinquent principal and defaults to be covered before interest is paid on the class M notes. 

The buyer of this swaption has the right, one year from now, to enter into a 3-year swap as the fixed-rate payer, paying 4% p.a. against receiving 3-month EURIBOR, on a notional principal of 10 million. If 3-year swap rates on 29 March 20X4 were, say, 4.5%, it would be worthwhile for the owner to exercise the swaption, paying a fixed rate of only 4% when the market rate was 4.5%. 

A neat way round this problem is to reduce the fixed rate of the swap below the market s fair rate of 3.00%, and have the swap counterparty make an up-front payment to the investor to restore parity. Reducing the swap rate also means that the strike rate of the payer s swaption must also be reduced in line, which increases the up-front premium payable. 

If 2-year swap rates are higher than 2.60% in three years time, the investor would cancel the original swap, and either enter into another swap at a better rate, or enjoy the higher floating rate from the FRN. 

In an interest rate swap, the counterparties agree to exchange periodic interest payments. The euro amount of the interest payments exchanged is based on the notional principal. In the most common type of swap, there is a fixed-rate payer and a fixed-rate receiver. The convention for quoting swap rates is that a swap dealer sets the floating rate equal to the reference rate and then quotes the fixed rate that will apply. 

In the previous section we described in general terms the payments by the fixed-rate payer and fixed-rate receiver but we did not give any details. That is, we explained that if the swap rate is 6% and the notional amount is 100 million, then the fixed-rate payment will be 6 million for the year and the payment is then adjusted based on the frequency of settlement. So, if settlement is semiannual, the payment is 3 million. If it is quarterly, it is 1.5 million. Similarly, the floating-rate payment would be found by multiplying the reference rate by the notional amount and then scaled based on the frequency of settlement. 

At the inception of the swap, the terms of the swap will be such that the present value of the floating-rate payments is equal to the present value of the fixed-rate payments. That is, the value of the swap is equal to zero at its inception. This is the fundamental principle in determining the swap rate (i.e., the fixed rate that the fixed-rate payer will make). 

It is the same equation as for determining the floating-rate payment except that the swap rate is used instead of the reference rate (3-month EURIBOR in our illustration). 

For example, suppose that the swap rate is 4.98% and the quarter has 90 days. Then the fixed-rate payment for the quarter is... 

Exhibit 19.4 shows the fixed-rate payments based on different assumed values for the swap rate. The first three columns of the exhibit show the same information as in Exhibit 19.3—the beginning and end of the quarter and the number of days in the quarter. Column (4) simply uses the notation for the period. That is, period 1 means the end of the first quarter, period 2 means the end of the second qnarter, and so on. The other columns of the exhibit show the payments for each assumed swap rate. 

Now that we know how to calcnlate the payments for the fixed rate and floating-rate sides of a swap, where the reference rate is 3-month EURIBOR given (1) the cnrrent valne for 3-month EURIBOR (2) the expected 3-month EURIBOR from the EURIBOR futures contract and (3) the assnmed swap rate, we can demonstrate how to compnte the swap rate. 

At the initiation of an interest rate swap, the counterparties are agreeing to exchange future payments and no upfront payments by either party are made. This means that the swap terms must be such that the present value of the payments to be made by the counterparties must be at least equal to the present value of the payments that will be received. In fact, to eliminate arbitrage opportunities, the present value of the payments made by a party will be equal to the present value of the payments received by that same party. The equivalence (or no arbitrage) of the present value of the payments is the key principle in calculating the swap rate. 

EXHIBIT 19.4 Fixed-Rate Payments for Several Assumed Swap Rates... 

Quarter Starts Quarter Ends Number of Days in Quarter Period = End of Quarter Eixed-Rate Payment If Swap Rate Is Assumed to Be ... 

The fixed-rate payer will require that the present value of the fixed-rate payments that must be made based on the swap rate not exceed the 14,052,917 payments to be received from the floating-rate payments. The fixed-rate receiver will require that the present value of the fixed-rate payments to be received is at least as great as the 14,052,917 that must be paid. This means that both parties will require a present value for the fixed-rate payments to be 14,052,917. If that is the case, the present value of the fixed-rate payments is equal to the present value of the floating-rate payments and therefore the value of the swap is zero for both parties at the inception of the swap. The interest rates that should be used to compute the present value of the fixed-rate payments are the same interest rates as those used to discount the floating-rate payments. 

To show how to compute the swap rate, we begin with the basic relationship for no arbitrage to exist ... 

Let s apply the formula to determine the swap rate for our 3-year swap. Exhibit 19.7 shows the calculation of the denominator of the formula. The forward discount factor for each period shown in Column (5) is obtained from Column (4) of Exhibit 19.6. The sum of the last column in Exhibit 19.7 shows that the denominator of the swap rate formula is 281,764,282. We know from Exhibit 19.6 that the present value of the floating-rate payments is 14,052,917. Therefore, the swap rate is... 

EXHIBIT 19.7 Calculating the Denominator for the Swap Rate Formula... 

Given the swap rate, the swap spread can be determined. For example, since this is a 3-year swap, the convention is to use the 3-year rate on the euro benchmark yield curve. If the yield on that issue is 4.5875%, the swap spread is 40 basis points (4.9875% - 4.5875%). 

The calculation of the swap rate for all swaps follows the same principle equating the present value of the fixed-rate payments to that of the floating-rate payments. 

To illustrate this, consider the 3-year swap used to demonstrate how to calculate the swap rate. Suppose that one year later, interest rates change as shown in Columns (4) and (6) in Exhibit 19.8. In Colnmn (4) shows the current 3-month EURIBOR. In Column (5) are the EURIBOR futures price for each period. These rates are used to compute the forward rates in Column (6). Note that the interest rates have increased one year later since the rates in Exhibit 19.8 are greater than those in Exhibit 19.3. As in Exhibit 19.3, the current 3-month EURIBOR and the forward rates are used to compute the floating-rate payments. These payments are shown in Column (8) of Exhibit 19.8. 

In Exhibit 19.10 the forward discount factor (from Exhibit 19.9) and the floating-rate payments (from Exhibit 19.8) are shown. The fixed-rate payments need not be recomputed. They are the payments shown in Column (8) of Exhibit 19.4. This is the fixed-rate payments for the swap rate of 4.9875% and is reproduced in Exhibit 19.10. Now the two payment streams must be discounted using the new forward discount factors. As shown at the bottom of Exhibit 19.10, the two present values are as follows ... 

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