Interest rate swaps pricing

The market approach to CDS pricing adopts the same no-arbitrage concept that is used in interest rate swap pricing. This states that, at inception ... 

Z-spread is an alternative spread measure to the ASW spread. This type of spread uses the zero-coupon yield curve to calculate the spread, in which in this case is assimilated to the interest-rate swap curve. Z-spread represents the spread needful in order to obtain the equivalence between the present value of the bond s cash flows and its current market price. However, conversely to the ASW spread, the Z-spread is a constant measme. 

Consequently, interest rate swaps can be viewed as a package of more basic interest rate derivative instruments—forwards. The pricing of an interest rate swap will then depend on the price of a package of forward contracts with the same settlement dates in which the underlying for the forward contract is the same reference rate. 

The products discussed include interest rate swaps, options, and credit derivatives. There is also a chapter on the theory behind forward and fiimres pricing, with a case smdy featuring the price history and implied repo rate for the CBOT long bond future. 

This chapter discusses the uses of interest rate swaps, including as a hedging tool, from the point of view of bond-market participants. The discussion touches on pricing, valuation, and credit risk, but for complete... 

FIGURE 7.5 Pricing a Plain Vanilla Interest Rate Swap ... 

It is not surprising that the net present value is zero. The zero-coupon curve is used to derive the discount factors that are then used to derive the forward rates that are used to determine the swap rate. As with any financial instrument, the fair value is its break-even price or hedge cost. The bank that is pricing this swap could hedge it with a series of FRAs transacted at the forward rates shown. This method is used to price any interest rate swap, even exotic ones. 

Asset-swap pricing is commonly applied to credit-default swaps, especially by risk management departments seeking to price the transactions held on credit traders book. A par asset swap typically combines an interest rate swap with the sale of an asset, such as a fixed-rate corporate bond, at par and with no interest accrued. The coupon on the bond is paid in return for LIBOR plus, if necessary, a spread, known as the asset-swap spread. This spread is the price of the asset swap. It is a function of the credit risk of the underlying asset. That makes it suitable as the basis for the price payable on a credit default swap written on that asset. 

The other major use by banks of credit derivatives is as a product offering for clients. The CDS market has developed exactly as the market did in interest rate swaps, with banks offering two-way prices... 

On the no-arbitrage principle, which is the same approach used to price interest rate swaps, for a CDS to be fairly priced, the expected value of the premium stream must equal the expected value of the default payment. 

An asset swap is a package that combines an interest-rate swap with a cash bond, the effect of the combined package being able to transform the interest-rate basis of the bond. Typically, a fixed-rate bond will be combined with an interest-rate swap in which the bond holder pays fixed coupon and received floating coupon. The floating coupon will be a spread over LIBOR (see Choudhry et al. 2001). This spread is the asset-swap spread and is a function of the credit risk of the bond over and above interbank credit risk. Asset swaps may be transacted at par or at the bond s market price, usually par. This means that the asset swap value is made up of the difference between the bond s market price and par, as well as the difference between the bond coupon and the swap flxed rate. 

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. ... 

A credit default swap (CDS) price provides fundamental credit risk information of a specific reference entity or asset. As explained before, asset swaps are used to transform the cash flows of a corporate bond for interest rate hedging purpose. Since the asset swaps are priced at a spread over the interbank rate, the ASW spread is the credit risk of the same one. However, market evidence shows that credit default swaps trade at a different level to asset swaps due to technical... 

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. 

Jamshidian, F., 1997. LIBOR and swap market models and measures. Finance Stochast. 1,293-330. Jarrow, R., Madan, D., 1991. Option Pricing Using the Term Structure of Interest Rates to Hedge Systematic Discontinuities in Asset Returns Working Paper. Cornell University, Ithaca, NY,... 

The two previous chapters introduced and described a fractiOTi of the most important research into interest-rate models that has been carried out since the first model, presented by Oldrich Vasicek, appeared in 1977. These models can be used to price derivative seciuities, and equitibrium models can be used to assess fair value in the bond market. Before this can take place however, a model must be fitted to the yield curve, or calibrated In practice, this is carried out in two ways the most popular approach involves calibrating the model against market interest rates given by instruments such as cash Libor deposits, futures, swaps and bonds. The alternative method is to model the yield curve from the market rates and then calibrate the model to this fitted yield curve. The first approach is common when using, for example extended Vasicek... 

In other words, if market interest rates rise, the mark-to-market (mtm) value of a receive-fixed swap will be increasing as discounting rates rise. In turn, this means that the break-even rate of the swap moves lower hence a market making swap bank will require a lower fixed rate if it is to price the swap correctly as discounting rates rise. 

The minimum interest rate that an investor should require is the yield available in the marketplace on a default-free cash flow. For bonds whose cash flows are denominated in euros, yields on European government securities serve as benchmarks for default-free interest rates. In some European countries, the swap curve serves as a benchmark for pricing spread product (e.g., corporate bonds). For now, we can think of the minimum interest rate that investors require as the yield on a comparable maturity benchmark security. 

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. 

As explained in chapter 3, zero-coupon, or spot, rates are true interest rates for their particular terms to maturity. In zero-coupon swap pricing, a bank views every swap, even the most complex, as a series of cash flows. The zero-coupon rate for the term from the present to a cash flows payment date can be used to derive the present value of the cash flow. The sum of these present values is the value of the swap. 

A more accurate approach m ht be the one used to price interest tate swaps to calculate the present values of future cash flows usit discount tates determined by the markets view on where interest rates will be at those points. These expected rates ate known as forward interest rates. Forward rates, however, are implied, and a YTM derived using them is as speculative as one calculated using the conventional formula. This is because the real market interest rate at any time is invariably different from the one implied earlier in the forward markets. So a YTM calculation made using forward rates would not equal the yield actually realized either. The zero-coupon rate, it will be demonstrated later, is the true interest tate for any term to maturity. Still, despite the limitations imposed by its underlying assumptions, the YTM is the main measure of return used in the markets. 

The zero-coupon curve is used in the asset swap valuation. This curve is derived from the swap curve, so it is the implied zero-coupon curve. The asset swap spread is the spread that equates the difference between the present value of the bond s cash flows, calculated using the swap zero rates, and the market price of the bond. This spread is a function of the bond s market price and yield, its cash flows, and the implied zero-coupon interest rates. ... 

Further, if there were a variety of bonds of a particular issuer outstanding, with different maturities, a term structure of hazard rates could be constructed—which in turn could be used to price default swaps of any maturity. By reducing everything to the hazard rate X, we are able to calculate correctly the prices of different instruments regardless of their interest or premium payment frequencies and daycount conventions. Similarly each instrument s mechanics are stripped away (e.g., a default swap versus a bond) to reveal the true hazard rate. 

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