Quantitative finance, also called mathematical finance, is the application of mathematics to finance. The techniques include:
Continuous vs discrete time finance
Financial economics and quantitative finance use both continuous time and discrete time approaches. Continuous time finance assumes exactly what it says. Continuous time models usually assume that prices are continuous (i.e. ignore ticks).
The majority of commonly used closed form results (i.e. equations) such as the Black-Scholes formula are derived using continuous time techniques: bot oftem have discrete time equivalents. Discrete time models usually require less maths to understand the models and to derive results.
Limitations of quantitative finance
The issues are those of model risk in general, but two problems are worth discussing because they become acute when complex techniques are used. One that of incorrectly estimating parameters, the other is the use of incorrect distributions.
The underlying problem is that lack of data. The period over which historical data is collected may not cover unusual but important periods such as market crashes. This leads to the use of distributions that are not sufficiently fat tailed, because they appear to give accurate results even when back tested. In addition, the lack of large but unusual variation can lead to the under-estimation of parameters such as standard deviation.
Some commentators claim that the common approaches are entirely flawed because of the risk of black swan events (which no amount of historical data would contain).
Although these criticisms most clearly apply to risk management techniques such as value-at-risk, valuations reflect risk, so they also have implications for valuation models.
Apart from the risks implicit in the process of modelling, complex models are more prone to manipulation because their complexity means that there are more, and less clear, choices in how they are implemented, and they are therefore more dependent of the judgment of the implementor. This has become evident in the aftermath of the credit crunch, as many banks and fund managers appear to have pressured quants to avoid models that produced high (as it turned out, all too accurate) estimates of risk.