Risk Latte - Working Miracles with the Differential Operator - II

Ornstein-Uhlenbeck Process in Interest Rate Models

Team Latte
18th January 2015

Deutsche Bank quants, Marcus Overhaus, Hans Buehler, Ana Bermudez, Andrew Ferraris, Christopher Jordinson and Aziz Lamnouar in their book Equity Hybrid Derivative, have shown that within the context of short rate modeling if the short rate (one period annualized interest rate), , is a function of the variable and its mean, such that , then the short rate can be expressed as a generalized mean reverting Ornstein-Uhlenbeck (OU) process given by:

This generalized OU process yields most of the well-known single factor short rate models. For example, if, , then we recover the Vasicek or the Hull-White model; if we let the short rate take exponential form of the type then we recover the Black-Karasinkski model. And, for such an exponential form for short rate, if we make then we retrieve the Black-Derman-Toy model.

Further, if we want to have a better grip on the functional relationship between the short rate and its volatility then a parameter can be introduced by interpolating between a normal and a lognormal model so that the short rate is expressed as:

Here, in the limit when , we recover the Hull-White model and when , we retrieve the Black-Karasinski model.

Reference:

  1. Overhaus, Marcus, Hans Buehler, et al, Equity Hybrid Derivatives, John Wiley & Sons, 2007.
  2. Bhattacharya, Rahul, The Book of Greeks, CFE Publishing, (Unpublished Draft) 2013

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