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Unformatted text preview: 1 15 ContinuousTime Limits This chapter discusses the computer implementation of the interest rate option models developed in the previous chapters. As a discretetime model, its approximation to reality is good when the number of periods ( ) is large. In this case, the discretetime model is approximating the continuous trading limit. In fact, for purposes of empirical estimation, it is convenient to reparameterize the discretetime model in terms of its continuoustime limit. The actual implementation of computer code is then done under this reparameterization. The primary purpose of this chapter is to study this reparameterization and the resulting continuoustime limit. A secondary purpose is to demonstrate how to construct arbitragefree zerocurve evolutions such as those used in the examples of the previous chapters. 2 A Motivation This section discusses the intuition behind the construction of the discrete time approximation to the continuoustime limit economy. To parameterize the forward rate process in terms of its continuous limit, we need to change the time scale in the discretetime model. As it is currently constructed, there are time periods t = , 1 , 2 ,..., . These time periods are arbitrarily specified. 3 In order to take limits, let us fix a future date (say, January 1, 2030), and divide the time horizon 0 to into subperiods of equal length . In terms of calendar time, the discrete periods 0 , 1 , 2 ,..., correspond to the dates 0, , 2 , 3 ,..., . We are interested in studying the various discretetime economies when the number of trading dates becomes large (i.e., ) or, equivalently, when the time between trades becomes small (i.e., ). 4 The discretetime forward rate is denoted ) T , t ( f . Recall that this represents one plus the percentage forward rate at time t for the future time period [T,T+ ]. The continuously compounded (continuoustime) forward rate ) T , t ( f ~ corresponds to that rate such that ) T , t ( f ~ e ) T , t ( f . It is that rate, compounded continuously for units of time that equals the discrete rate. Note that this expression implies that ) T , t ( f ~ is a percentage . More formally, ( ) T , t f log lim ) T , t ( f ~ . 5 In constructing the continuoustime economy, we are concerned with changes in continuously compounded forward rates, i. e. . ) T , t ( f log ) T , t ( f log ) T , t ( f ~ ) T , t ( f ~ + + From the continuoustime perspective, the evolution of observed zerocoupon bond prices and forward rates are generated by a continuous time empirical economy with parameters (i) * (t,T) , the expected change in the continuously compounded forward rates per unit time, and (ii) (t,T) , the standard deviation of changes in the continuously compounded forward rates per unit time....
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This note was uploaded on 10/31/2010 for the course NBA 5550 taught by Professor Jarrow,robert during the Fall '08 term at Cornell University (Engineering School).
 Fall '08
 JARROW,ROBERT

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