Mathematical finance is a field of applied mathematics, concerned with mathematical modeling of financial markets.
Derivative pricing: determine the fair price of a given security in terms of more liquid securities whose price is determined by the law of supply and demand.
ℚ vs ℙ
ℚ: risk-neutral probability (or arbitrage-pricing probability)
ℙ: actual (or actuarial) probability,
TODO
In a risk-neutral world, all invested assets (securities) are assumed to earn the same expected rate of return, the risk-free rate, regardless of the risks inherent in the specific invested asset.