Lecture_15 - 1 15 Continuous-Time Limits This chapter...

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Unformatted text preview: 1 15 Continuous-Time Limits This chapter discusses the computer implementation of the interest rate option models developed in the previous chapters. As a discrete-time model, its approximation to reality is good when the number of periods ( ) is large. In this case, the discrete-time model is approximating the continuous trading limit. In fact, for purposes of empirical estimation, it is convenient to reparameterize the discrete-time model in terms of its continuous-time 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 continuous-time limit. A secondary purpose is to demonstrate how to construct arbitrage-free zero-curve 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 continuous-time limit economy. To parameterize the forward rate process in terms of its continuous limit, we need to change the time scale in the discrete-time 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 discrete-time economies when the number of trading dates becomes large (i.e., ) or, equivalently, when the time between trades becomes small (i.e., ). 4 The discrete-time 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 (continuous-time) 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 continuous-time 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 continuous-time perspective, the evolution of observed zero-coupon 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).

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Lecture_15 - 1 15 Continuous-Time Limits This chapter...

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