Generally speaking, corporate finance is to study how a company should finance (financing projects) and how to invest (making money by investing in projects) from the company's point of view. Financing includes issuing stocks and bonds. Investment includes the selection of projects, mainly using NPV and other methods. The project I am talking about here is quite broad, which can include the operation of a production line in a factory or the drilling of oil wells.
Asset pricing usually studies whether the pricing of foreign exchange, stocks, bonds and other markets is reasonable and what is the reasonable price. Once the pricing is unreasonable, you take advantage of the arbitrage opportunity.
At present, most popular financial mathematics focuses on asset pricing, especially the financial derivatives market. Reasonable pricing of emerging financial derivatives through various mathematical models. This is different from the traditional finance mentioned above, because traditional finance does not use so many mathematical models. Derivative market is actually the expansion or extension of traditional foreign exchange, stocks, bonds and other markets.