Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics. Patrick Muldowney

      Observe that the number of shares held at any time depends on whether the share price has moved up or down. So , , is a deterministic function of ; and the value of varies randomly because varies randomly.

      The same applies to the values of , including the terminal value , or with . Table 2.5 gives the respective process sample paths for processes, where the underlying share price process follows sequence DUUU (sample number 2 in Table 2.4).

      Regarding notation, the symbols , (and so on) are used here, in contrast to symbols etc. which were used in discussion of stochastic calculus in Chapter 1 . In the latter, the emphasis was on the classical rigorous theory in which random variables are measurable functions, and this is signalled by using instead of , etc.

      Where (rather than etc.) is used, the purpose is to indicate the “naive” or “natural” outlook which sees random variability, not in terms of abstract mathematical measurable sets and functions, but in terms of actual occurrences such as tossing a coin, or such as the unpredictable rise and fall of prices.

      A mathematically rigorous approach to random variation can be squarely based on the latter view, and in due course this will provide mathematical justification for notation etc.

      Table 2.5 describes two out of a possible total of sixteen outcomes, or sample paths, for each of the processes involved. But the tables do not examine the probabilities of the various outcomes. So Table 2.4, for instance, does not really shed much light on how the investment policy of the portfolio holder (shareholder) is capable of performing. The alternative outcomes of the policy are displayed in Table 2.4, but on its own the list of outcomes does not say whether a gain of wealth is more likely than an overall loss.

What if, for instance, we wish to determine the expected overall gain in the value of the shareholding at the end of four days? With , what is the value⁴ of ?

      This can be answered directly as follows.

       Suppose the different possible amounts of total or net shareholding gain are known. Two of these, and , are calculated above. There are 16 possible sample paths for the underlying process corresponding to the 16 permutations of U, D. So, allowing for duplication of values, there are at most 14 other values for total shareholding gains.

       The probability of each of the 16 values of is the same as the probability of the corresponding underlying sample path (or ). It is assumed that the probability of a U or D transition is 0.5 in each case. If it is further assumed that the transitions are independent, then the probability of each of the 16 sample paths is , or one sixteenth. This is then the probability of each of 16 outcomes for total shareholding gain, including duplicated values.

      The 16 values for can be easily calculated, as in Table 2.5 above. In fact, the 16 outcomes for net wealth (shareholding value) gain are

      Since each of the transition sequences

      has equal probability , each of the 16 values (including duplicates) for has probability , or one sixteenth (due to the assumption of independence). Therefore, when all the details are fully calculated out,

(2.6)