In the equity asset class, the underlying is the price of a company stock. For instance, the current price of one share of Vodafone PLC (VOD.L) as quoted in the London Stock Exchange (www.londonstockexchange.com) at some particular time. The price could be £2.32 and the time could be 11:33:24 on May 13, 2013.
In mathematical terms, thus, the price of a stock can be represented as a scalar function of the current time t. We will denote this function as S(t). Note that in technical terms, S(t) is a time series, which even though apparently continuous (with C[0] continuity), is in reality discontinuous (subject to jumps). In addition, it is not a well-behaved function, that is, its first derivative does not exist.
We are going to model S(t) as an stochastic variable. And all the constructions that we build around this value, such as the value of the payoff H(S_t) will be in consequence stochastic functions. In this situation, we are required not to use the standard tools of calculus (such...