Time-Continuous Model

Trading on a stock market is obviously a discrete process, as it consists of single transactions performed at distinct times. There are, however, so many transactions in such a high frequency that it is for many applications better to model them in a time-continuous setting, i.e., to assume that they take place at all times. In this chapter we will provide a short introduction to time-continuous models. An important difference will be that prices are exogenously given. In particular, we will derive the famous Black-Scholes model Black-Scholes model for asset pricing as it has been introduced by Fischer Black and Myron Scholes (J Pol Econ 81(3):637–654, 1973) and by Robert C. Merton (Econometrica 41(5):867–887, 1973). In 1997, the Nobel prize has been awarded for this work. The importance of this model can not be overemphasized, as the Royal Swedish Academy put it: Their innovative work … has provided us with completely new ways of dealing with financial risk, both in theory and in practice. Their method has contributed substantially to the rapid growth of markets for derivatives in the last two decades. In fact, their formula is probably the most used formula on banks and stock exchanges worldwide even today. In this chapter we will derive this celebrated formula, but we will not reduce all to one equation. There is a whole theory behind this result and we will see how this theory can be used to solve many more problems. We will also see what limitations the classical theory still has and sketch some ways how to overcome them.