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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). We’ll 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 )Δ represents the amount of interest earned on average by all investors over a short period of time Δ. Some examples of models Example: a very simple model would be dr ( t ) = σr ( t ) dB ( t ) which means that for a small amount of time Δ, we have that Δ r ( t ) := r ( t + Δ)- r ( t ) = σr ( t )( B ( t + Δ)- B ( t )) . We can use the properties of BM, stating that B ( t + Δ)- B ( t ) ∼ N (0 , Δ) , to conclude that Δ r ( t ) r ( t ) ∼ N (0 ,σ 2 Δ) . We can show that, in turn, this implies we have r ( t ) = r (0) e- σ 2 t/ 2+ σB ( 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 doesn’t 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 won’t develop here. Example: Vasicek model. This model is described by dr ( t ) = a ( b- r ( t )) dt + σB ( t ) . Therefore, Δ r ( t ) = a ( b- r ( t ))Δ + σZ where Z ∼ N (0 , Δ) 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 ) + σe- 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
- Mathematical finance, Monte Carlo methods in finance, Short rate model