Interest rates derivatives

In this thesis we derived new methods for the pricing of fixed income derivatives, especially for zero-coupon bond options (caps/floor) and coupon bond options (swaptions). These options are the most widely traded interest rate derivatives. In general caps/floors can be seen as a portfolio of zero-coupon bond options, whereas a swaption effectively equals an option on a coupon bond (see chapter (2)). The market of these LIBOR-based interest rate derivatives is tremendous (more than 10 trillion USD in notional value) and therefore accurate and efficient pricing methods are of enormous practical importance. 

Chako G, Das S (2002) Pricing Interest Rate Derivatives A General Approach. The Review of Financial Studies 15 195-241. 

Heiddari M, Wu L (2003a) Are Interest Rate Derivatives Spanned by the Term Structure of Interest Rates . Journal of Fixed Income 13 75-86. 

Hull J, White A (1990) Pricing interest rate derivative securties. Review of Financial Studies 3 573-592. 

Li H, Zhao F (2006) Unspanned Stochastic Volatility Evidence from Hedging Interest Rate Derivatives. Journal of Finance 61 341-378. 

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. ... 

It is possible to generalise Ito s formula in order to produce a multi-dimensional formula, which can then be used to construct a model to price interest-rate derivatives or other asset-class options where there is more than one variable. To do this, we generahse the formula to apply to situations where the d5mamic function/() is dependent on more than one Ito process, each expressed as a standard Brownian motion. 

An interest-rate model provides a description of the dynamic process by which rates change over time, in terms of a statistical construct, as well as a means by which interest-rate derivatives such as options can be priced. It is often the practical implementation of the model that dictates which type is used, rather than mathematical neatness or more realistic assumptions. An excellent categorisation is given in James and Webber (2000), who list models as being one of the following types ... 

Hull, J., White, A., 1990. Pricing interest-rate derivative securities. Rev. Financ. Stud. 3, 573-592. James, J., Webber, N., 2000. Interest Rate Modelling. Wiley, Chichester. 

EXHIBIT 17.14 Growth of OTC versus Exchange-Traded Interest Rate Derivatives Markets... 

Unfortunately, without a central clearing house to monitor and record all transactions, it is difficult to obtain reliable statistics for the OTC interest rate options market. However, the BIS conducts regular surveys of the markets and publishes a breakdown of notional amounts outstanding by currency (but not by country) of all interest rate derivatives (including swaps, ERAs, futures, as well as options) across all markets (OTC as well as exchange-traded). A summary of this is shown in Exhibit 17.16, from which it can be seen that the size of the euro-denominated market now virtually matches that of the US dollar, signalling the increasing importance of the European interest rate derivatives market. 

EXHIBIT 17.16 Currency Composition of Interest Rate Derivatives... 

A cap is an interest rate derivative offering protection against unexpected fluctuations in short-term interest rates, but over an extended period of time. An example will make this clearer. 

In nearly all these cases, the structured product can be engineered using a combination of a straightforward fixed-rate bond and one or more interest rate derivatives. Examples follow. 

In addition to using vanilla interest rate options to create the structured products discussed in the previous section, banks can also create structured interest rate derivatives. These can be tailored to meet client needs and include the products discussed in the following paragraphs. In each case, we illustrate the structure by reference to an interest rate cap, but the same principles apply equally well to floors and collars. 

Althongh the sample option price is easy to read and interpret in respect of this screen, there is a mass of academic and practitioner research literatnre that provides a platform from which bond option prices in general can be calcnlated with integrity. The literature on modelling interest rate derivatives in this arena is freqnently divided into one-, two-factor, or mnltifactor, models. 

John C. Hull and Alan White, Pricing Interest Rate Derivative Securities, Review of Financial Studies 3, no. 5 (1990), pp. 573-592. 

Hull and White, Pricing Interest Rate Derivative Securities. ... 

Klaus Sandmann and Dieter Sondermann, A Term Structure Model and the Pricing of Interest Rate Derivatives, Review of Futures Markets 12, no. 2 (1993), pp. 391 23. 

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 swap curve depicts the relationship between the term structure and swap rates. The swap curve consists of observed market interest rates, derived from market instruments that represent the most liquid and dominant instruments for their respective time horizons, bootstrapped and combined using an interpolation algorithm. This section describes a complete methodology for the construction of the swap term structure. 

The value of the volatility parameter is user-specified—that is, it is set at a value that the user feels most accurately describes the current interest rate environment. The value used is often the volatility implied by the market price of interest rate derivatives such as caps and floors. 

Selecting the appropriate term-structure model is more of an art than a science, depending on the particular application involved and the users individual requirements. The Ho-Lee and BDT versions, for example, are arbitrage, or arbitrage-free, models, which means that they are designed to match the current term structure. With such models—assuming, of course, that they specify the evolution of the short rate correctly—the law of noarbitrage can be used to determine the price of interest rate derivatives. 

Equilibrium interest rate models also exist. These make the same assumptions about the dynamics of the short rate as arbitrage models do, but they are not designed to match the current term structure. The prices of zero-coupon bonds derived using such models, therefore, do not match prices seen in the market. This means that the prices of bonds and interest rate derivatives are not given purely by the short-rate process. In brief, arbitrage models take as a given the current yield curve described by the... 

This part also considers the primary fixed-income derivative instruments. These are not securities in the cash markets and are fixed-income derivatives (or interest rate derivatives) in the synthetic markets. 

As explained in chapter 2, a bond s modified or Macaulay s duration is the average time to receipt of its cash flows, weighted according to their present values. To compute a mortgage-backed bond s duration, it is necessary to project its cash flows using an assumed prepayment rate. These projections, together with the bond price and the periodic interest rate, derived from the yield, may then be used to arrive at the bond s periodic duration, which is divided by twelve (or four, in the case of a bond that pays quarterly) to arrive at its duration in years. 

Where can these careers lead Take the example of one engineer who worked in the energy industry building gas stations. He was hired as a project manager in derivative operations. An opportunity to manage a team of analysts supporting interest rate derivatives client confirmations arose as a result of previous successful project initiatives. That role soon ejq)anded into credit derivatives, which has been a continuously... 

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