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Unformatted text preview: ACTSC 445: Asset-Liability Management Fall 2008 Department of Statistics and Actuarial Science, University of Waterloo Unit 8 (Part II) Continuous-time interest rate models References (recommended readings): Chap. 28 of Hull. Introduction The binomial model is simple to use, but not very realistic unless each time-step represents a small enough amount of time. But then there are too many nodes and it is likely to become too cumbersome from a computational point of view. Rather than taking very small time steps in a tree, it is better to work with a continuous-time model , where we described the behavior of r ( t ) = short rate at time t for all t 0. Continuous-time models require the use of the Brownian motion (BM). Well be avoiding getting into the foundations of BM and will simply attempt to learn how to use these models. The models presented here are all risk-neutral models, which means r ( t ) represents the amount of interest earned on average by all investors over a short period of time . Some examples of models Example: a very simple model would be dr ( t ) = r ( t ) dB ( t ) which means that for a small amount of time , we have that r ( t ) := r ( t + )- r ( t ) = r ( t )( B ( t + )- B ( t )) . We can use the properties of BM, stating that B ( t + )- B ( t ) N (0 , ) , to conclude that r ( t ) r ( t ) N (0 , 2 ) . We can show that, in turn, this implies we have r ( t ) = r (0) e- 2 t/ 2+ B ( t ) 1 where, again by the properties of the BM, B ( t ) N (0 ,t ). Hence for each t , r ( t ) has a lognormal distribution. This model is a special case of the Rendleman&Bartter model. It is an equilibrium model, and thus the prices for zero-coupon bonds returned by this model may be different from market prices. It is also called a lognormal model, since, as we just pointed out, for this model r ( t ) has a lognormal distribution. A drawback of this model is that it doesnt include a mean-reversion feature, which is believed to be an important property that interest-rate models should have. Question: how do we get the price of a zero-coupon bond for a continuous-time model (came up when we said above model was an equilibrium model)? Answer: use the formula P (0 ,t ) = E e- R t r ( s ) ds , which is the expectation of the discounted value of $1 paid at time t . To compute this expectation, we need mathematical tools that we wont develop here. Example: Vasicek model. This model is described by dr ( t ) = a ( b- r ( t )) dt + B ( t ) . Therefore, r ( t ) = a ( b- r ( t )) + Z where Z N (0 , ) and the term ( b- r ( t )) captures the mean-reversion feature. For this model, we can show that r ( t ) = r (0) e- at + b (1- e- at ) + e- at Z t a as dB ( s ) and therefore E( r ( t )) = r (0) e- at + b (1- e- at ) b as t ....
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- Fall '09