Lecture_07

Lecture_07 - 7 Bond Trading Strategies An Example A...

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1 7 Bond Trading Strategies – An Example A Motivation This chapter shows how to use the concepts of trading strategies, arbitrage opportunities and complete markets to investigate mispricings within the yield curve. Taken as given are the stochastic processes for a few zero-coupon bonds, the stochastic process for the spot rate, and the assumption that there are no arbitrage opportunities. The purpose is to find the arbitrage-free prices of all the remaining zero-coupon bonds.
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2 Figure 7.1: A Graphical Representation of “Arbitraging the Zero-coupon Bond Price Curve” Time (T) Prices P(0,T) |||| 01234 1 X F M F=M X X given price F fair price M market price
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3 B Method 1: Synthetic Construction The first method concentrates on a technique for constructing the zero-coupon bond synthetically, using the money market account and the 4-period zero.
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4 Figure 7.2: An Example of a Four-Period Zero-Coupon Bond Price Process. The Value of the Money Market Account and the Spot Rates are Included on the Tree. Actual Probabilities Along Each Branch of the Tree. 1 1 1 1 1 1 1 1 4 1.054597 .985301 1.054597 .981381 1.059125 .982456 1.059125 .977778 1.062869 .983134 1.062869 .978637 1.068337 .979870 1.068337 .974502 3 1.037958 .967826 1.037958 .960529 1.042854 .962414 1.042854 .953877 1.02 .947497 1.02 .937148 2 1 0 time 3/4 1/4 3/4 1/4 3/4 1/4 3/4 1/4 3/4 1/4 3/4 1/4 3/4 1/4 B(0) = 1 P(0,4) = .923845 r(0) = 1.02 1.022406 1.016031 1.020393 1.019193 1.024436 1.017606
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5 1 An Arbitrage-free Evolution The first step in the procedure is to investigate whether the exogenous evolution as given in Figure 7.2 is arbitrage-free. We are taking the evolution in Figure 7.2 as given, and then from it, we will derive arbitrage-free prices for the 2- and 3-period zero-coupon bonds. The procedure will be flawed and nonsensical if the basis for the technique itself contains imbedded arbitrage opportunities. The check to determine if the given evolution is arbitrage-free proceeds node by node.
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6 First, consider ourselves standing at time 0. There are two securities available for trade at this date, the money market account and the 4-period zero-coupon bond. The money market account is riskless over the next time period earning 1.02. The 4-period zero-coupon bond is risky. The return on the 4-period zero-coupon bond in the up state is ( u(0,4) = .947497/.923845 = 1.025602) and in the down state it is ( d(0,4) = .937148/.923845 = 1.014400).
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7 In the up state it earns more than the money market account and in the down state it earns less. Hence, neither security dominates the other (in terms of returns). There would be no way to form a portfolio of these two securities with zero initial investment that didn’t have potential losses at time 1. Furthermore, any portfolio with an initial positive cash flow would have a negative cash flow at time 1 in at least one state. Thus, the fact that neither security dominates the other implies there are no arbitrage opportunities between time 0 and time 1. The converse of this statement is also true.
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This note was uploaded on 12/09/2008 for the course NBA 5550 taught by Professor Jarrow,robert during the Fall '08 term at Cornell.

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Lecture_07 - 7 Bond Trading Strategies An Example A...

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