We now, finally, have all the tools that we need for pricing and hedging in the continuous time world of Black and Scholes. We shall begin with the most basic setting, in which our market has just two securities: a cash bond and a risky asset whose price is modelled by a geometric Brownian motion.
In §5.1 we prove the Fundamental Theorem of Asset Pricing in this framework. In line with our analysis in the discrete world, this provides an explicit formula for the price of a derivative as the discounted expected payoff under the martingale measure. Just as in the discrete setting, we shall see that there are three steps to replication. In §5.2 we put this into action for European options. For simple calls and puts, the expectation that gives the price of the claim can be evaluated. We also obtain an explicit expression for the stock and bond holding in the replicating portfolio, via an application of the Feynman–Kac representation.
The rest of the book consists of increasing the complexity of the derivative contracts and of the market models. Before embarking on this programme, we relax the financial assumptions that we have made within the basic Black–Scholes framework. The risky asset that we have specified has a very simplistic financial side. We have assumed that it can be held without additional cost or benefit and that it can be freely traded at the quoted price.