Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues

8

The basis of stochastic calculus

8.1 STOCHASTIC PROCESSES

Stochastic is equivalent to random, hence the stochastic calculus develops rules of calculus to be applied if the problems to be handled are of a random (probabilistic) nature, in contrast with a deterministic one. As an example, unlike many physical phenomena (such as, for example, the trajectory of a bullet), the evolution of the prices or returns of financial products should intuitively not be considered as certain (deterministic). To be more realistic, their study should rather incorporate some random feature.

The deterministic or non-deterministic character of these financial products can be detected during the course of the time. It will thus concern forward products. The deterministic approach leads to the valuation of products such as vanilla swaps and futures, for which the forward value is obtained independently from the further evolution of their underlying instrument. The non-deterministic approach allows for taking into account a random evolution of the underlying spot instrument, which is necessarily the case for valuing products conditioned by such an evolution, that is, for options or any products presenting a conditional feature (for example, credit default swaps).

The evolution of the prices or returns of a financial instrument is to be represented by a mathematical model describing, at best, how prices or returns behave. It is important to distinguish between a forecasting model and an ex post – or ...