If you own the second edition of Arbitrage Theory in Continuous Time, I don't think owning the third edition will add substantial value. The two major chapters that were added are the martingale approach to optimal investment problems and optimal stopping theory. Apart from this, the book looks and reads like the second edition.
However, if you do not own the second edition, this book provides an excellent elementary treatment of asset pricing. The mathematics is quite reasonable, and does not require a substantial understanding of heavy math machinery from the reader. Many of the mathematical tools are explained in adequate detail in the text as well as in the Appendix. However, the reader should be comfortable with calculus and probability theory. The discussion on options is particularly good, especially the treatment of the binomial approach as well as Black-Scholes. However, this is not unique to this book.
I would say that this book would be a good supplement for students that are taking their intro level Ph.D. asset pricing course. In particular, I think this would be beneficial to those who would like to get a little bit more intuition than what they can get from standard Ph.D. level texts. Bjork's writing style may be helpful in that respect.