European option pricing
6.1 Contingent Claims
In this chapter we will focus on the determination of the fair price of a derivative, like options. We will assume geometric Brownian motion for asset price dynamics and pricing formulas of the Black and Scholes (1973) and Merton (1973) model.
Although we mainly discuss European call options we will discuss pricing of generic contingent claims which include call and put but also Asian options or options with barriers. Options of American type require a special attention and hence will be treated separately in Chapter 7.
In Chapter 8 we will consider deviations from the theory presented here. The main ingredient to be replaced in the Black and Scholes theory will be the model which describes assets prices to admit jumps or non-Gaussian returns.
We assume that there is a probability space , and a filtration . When not explicitly mentioned, all processes are adapted to the filtration .
A T-contingent claim or T-claim is a contract which pays to the holder a stochastic amount X at time T. The random variable X is