The field of quantitative finance is rich and interesting. When I mention the term "Financial Engineering" many people quip "Oh, now finance too has engineering..". Well, its aptly called that way because of the application of mathematical and engineering methods to problems in finance. But there is a lot of physics too. When I see Brownian Motion being applied to stock prices, there is a part of me thatsvery happy to use a tiny bit of the statistical physics that I learnt.
2005 is the world year of physics. Someone mentioned this is because, in 1905, Einstein published three most important papers in physics history - general relativity, photoelectric effect and brownian motion. Its surprising how a 100 years later brownian motion is applied in economics and finance to study stock prices like random motion of particles in a fluid. The famous black scholes model for option pricing uses quantum physics to make it simpler, unlike when it uses Ito's calculus.
Another important use of physics in finance is attributed to Feynman, for the sum-over-paths integral method (electromagnetic particles).
Econophysics is a relatively new branch, which lists a whole bunch of other applications of mathematical physics to finance. http://www.derivativesstrategy.com/ has some nice comic strips on trades, hedge funds and other aspects of finance.
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