Change of measure for the square-root process

The square-root process is used to model interest rates and volatility in financial mathematics. The pricing of derivatives involving that process often requires simulating it, since there are often no explicit formulas for prices. We study how a change of measure (CM) may improve those simulations. We compare with Andersen´s quadratic-exponential scheme (QE), which so far appears to be the most efficient technique to simulate the stochastic differential equation satisfied by the square-root process. An integer-dimension squared Bessel process, easy to simulate, is used to generate the law of the square-root process using a change of measure. The new method performs very well, and the two algorithms execute at similar speeds; however, CM is slower than QE if random number generation is taken into account, because CM requires more random numbers. The Radon-Nikodym derivative sometimes has a rather intriguing behavior, which is itself of interest. We propose an explanation.