After the groundbreaking contributions by Black, Scholes and Merton in the early 1970s, continuous time finance quickly grew into a large branch of financial economics. The area was strongly invigorated by the enormous proliferation of option trading and it received scientific recognition with the Nobel prize for Merton and Scholes in 1997. Sundaresan (2000) gives an overview of its developments and applications in various areas. Continuous time finance is an area where the mathematical approach can be fully exploited. Given securities prices on complete and arbitrage-free markets, the full force of mathematics can be brought to bear on the problem of how to price new securities. We have already developed the conceptual framework in discrete time, so we can introduce the continuous time techniques with minimal effort. We begin by having another look at the properties of stock returns. We then illustrate how probabilities can be transformed with the simple example of loading a die. Transformation is applied to stock returns, which results, in an equivalent martingale probability measure for stock returns. Pricing can then be done in the by now familiar way of discounting the risk-neutral expectation using the risk-free discount rate. The result is the celebrated Black and Scholes formula.

Preliminaries: stock returns and a die

So far we have mainly used simple, discretely compounded returns. In Black and Scholes’ option pricing we follow individual stocks in continuous time and this makes it necessary to use continuously compounded returns and to detail their properties.