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Unformatted text preview: Derivative Securities Fall 2007 Section 9 Notes by Robert V. Kohn, extended and improved by Steve Allen. Courant Institute of Mathematical Sciences. Interest-based instruments: bonds, forward rate agreements, and swaps. This section provides a fast introduction to the basic language of interest-based instruments, then introduces some specific, practically-important examples, including forward rate agreements and swaps. Most of this material can be found in Hull (chapters 4, 6, and 7). Steve Allens version of these notes starts with several pages of background about bonds and bond markets, including how the LIBOR (London Interbank Offering Rate) is set, why we can basically ignore credit risk when working with LIBOR rates, why its better to work with LIBOR rather than US Treasury rates, etc. I recommend reading that material, but I wont repeat it here. ************************ Bond prices and term structure. The time-value of money is expressed by the discount factor B ( t, T ) = value at time t of a dollar received at time T. This is, by its very definition, the price at time t of a zero-coupon bond which pays one dollar at time T . If interest rates are stochastic then B ( t, T ) will not be known until time t . Prior to time t it is random just as in our discussion of equities, a stock price s ( t ) or a forward price F ( t ) was random. Note however that B ( t, T ) is a function of two variables, the initiation time t and the maturity time T . Its dependence on T reflects the term structure of interest rates. (The forward prices in our prior discussions also had a term structure the forward price depends on the settlement date but the settlement date was usually held fixed. With interest rates, by contrast, well be considering many maturities simultaneously.) We usually take the convention that the present time is t = 0; thus what is observable now is B (0 , T ) for all T > 0. A central principle for dealing with the term structure of interest rates is to use the same discount factor B ( t, T ) for any cash flow that occurs at time T , regardless of what other cash flows it might be packaged with. For example, a cash flow occurring on June 15, 2012 will the discounted by the same factor whether it is a principal payment on a zero-coupon bond, a coupon payment on a 10 year bond, or a coupon payment on a 20 year bond. This principle follows from the law of one price; if it didnt hold we would be able to design arbitrage strategies, making money without taking any risk. There are several equivalent ways to represent the time-value of money. The yield y ( t, T ) is defined by B ( t, T ) = e- y ( t,T )( T- t ) ; it is the unique constant interest rate that would have the same effect as B ( t, T ) under continuous compounding. The term rate R ( t, T ) is defined by B ( t, T ) = 1 1 + R ( t, T )( T- t ) ; 1 it is the unique interest rate that would have the same effect as B ( t, T ) with no compounding....
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