Читать книгу Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - Patrick Muldowney - Страница 24

Suppose is time, measured in days. Suppose a share, or unit of stock, has value on day ; suppose is the number of shares held on day ; and suppose is the change in the value of the shareholding on day as a result of the change in share value from the previous day so . Let be the cumulative change in shareholding value at end of day , so . If share value and stockholding are subject to random variability, how is the gain (or loss) from the stockholding to be estimated?

Take initial value (at time ) of the share to be (or ), take the initial shareholding or number of shares owned to be (or ). Then, at end of day 1 (),

(2.1)

At end of day ,

(2.2)

After days,

(2.3)

If the time increments are reduced to arbitrarily small size (so represents number of “time ticks”—fractions of a second, say), with the meaning of the other variables adjusted accordingly, then

(2.4)

The latter expressions are Riemann sum estimates of (a Stieltjes‐type integral) whenever the latter exists.

Each of the expressions in (2.4) is sample value of a random variable

(2.5)

constructed from the random variables , , and . These notations symbolize—in a “naive” or “realistic” way—the stochastic integral of the process with respect to the process . In chapter 8 of [MTRV], symbols , or , or are used (in place of the symbol ) for various kinds of stochastic integral. In the context described here, would be the appropriate notation. (See (5.28) below.)

To illustrate the details of this basic stochastic integration, suppose the time increment is 1 day, so tracks the process over four days. Suppose the initial value of the share at the start of day 1 is . Suppose on each day the value of the share can change by or . That is, an “up” increment (U) or “down” increment (D). (Although the probabilities involved will not be used at this stage, in order to keep random variability in mind suppose that, at the end of each day, U occurs with probability and suppose D occurs with probability .)

Suppose initial stockholding at start of day 1 is , or 1 share, and suppose the shareholder buys an extra share whenever the share value increases (U), and otherwise keeps the same number of shares. So there are no circumstances in which shareholding is decreased. (It is easy to imagine that the investor would apply a less optimistic and more prudent share purchasing strategy. But for purpose of illustration some particular strategy must be chosen, and this one is easy to describe.)

The up (U) or down (D) changes in share price over four days are listed in Table 2.4. There are , , possible processes or histories, corresponding to the 16 possible permutations‐with‐repetition of the 2 symbols U and D, taken four at a time.

With and , the histories or processes of interest are prices ; holdings ; and total gains ; represented by

with representing the daily changes in the value of the stockholding. For , the ’tuple

is a sample path for the processes . Similarly for ’tuples , and processes , , respectively. One of the 16 underlying random transition sequences is no. 7 in Table 2.4:

With , , the corresponding sample paths

can be calculated for this particular sequence of share price U‐D transitions, using

for , as in Table 2.5.

Table 2.4 UD sample paths for processes

1.	U	U	U	U	10
2.	D	U	U	U	5
3.	U	D	U	U	4
4.	D	D	U	U	1
5.	U	U	D	U	3
6.	D	U	D	U	0
7.	U	D	D	U	‐1
8.	D	D	D	U	‐2
9.	U	U	U	D	2
10.	D	U	U	D	‐1
11.	U	D	U	D	‐2
12.	D	D	U	D	‐3
13.	U	U	D	D	‐3
14.	D	U	D	D	‐4
15.	U	D	D	D	‐5
16.	D	D	D	D	‐4

Table 2.5 Calculations for two UD sample paths for processes

s	0	1	2	3	4
		U	D	D	U
	10	11	10	9	10
	1	2	2	2	3
	0
		D	U	U	U
	10	9	10	11	12
	1	1	2	3	4
	0

For transition sample path number 7, UDDU, the overall gain in shareholding value is

where is a “negative gain” or net loss. With , this can be interpreted as the Stieltjes integral³

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)

the sum being taken over all 16 values (including duplicate values) of total gain .

When duplicate values are combined, there are 12 distinct outcomes for . Each of the duplicated outcomes has probability , while each of the other 8 distinct outcomes has probability .

To find the expected value of (or ) in accordance with the classical, rigorous mathematical theory of probability, it should be formulated in terms of a probability space , so

(2.7)

There are many ways in which a sample space can be constructed. One way is to let be the set of numbers consisting of the different values of (i.e. without duplicate values), of which there are 12, and let be the appropriate atomic probability measure on these 12 values. Letting be the identity function on , (or ) is measurable (trivially), and

because the integral in (2.7) reduces to the sum in (2.6.

Now suppose that, at times , the probability of an Up transition in is , while the probability of a Down transition in is :

and suppose, as before, that Up or Down transitions are independent of each other; so, for instance, the joint transition sequence U‐D‐D‐U (and the corresponding ) has probability

with similar probability calculations for each of the other 15 transition paths and their corresponding values (including duplicates, such as D‐U‐U‐U which also gives ).

The 16 outcomes (including replicated outcomes) for accumulated gain are the same as before, but because the probabilities are different, the expected net gain is now

(2.8)

Both calculations reduce to the same finite sum of terms. It is seen here that the new probability distribution, favouring Up transitions, produces an overall net gain in wealth through the policy of acquiring shares on an up‐tick, while not shedding shares on a down‐tick—the “optimistic” policy, in other words.

If the joint transition probabilities

are not independent, then, provided the dependencies between the various transitions and events are known, it is still possible to calculate all the relevant joint probabilities. But generally this is not so simple as the rule (of multiplying the component probabilities) that obtains when the joint occurrences are independent of each other.

A key step in the analysis is the construction of the probabilities for the values of the random variable (or ). The framework for this is as follows. Consider any subset of the sample space

(2.9)

whose elements are the different values which can be taken by the variable . For instance, , which is a member of the family of all subsets of .

Following through the logic of the classical theory, probability is defined on the family of measurable subsets of . A random variable is a real‐number‐valued, and ‐measurable, function

in this case, where the potential values are the numbers in the right‐most column of Table 2.4. The latter set is finite; and every finite subset, such as , is measurable. In fact, with sample space chosen in this way, is the identity function, since we have chosen so that its elements are the distinct values .

To find the probability of a set of ‐outcomes, such as , the classical theory requires that the corresponding set be found so that

gives the probability of outcomes as the corresponding probability in the sample space. Conveniently, in this case is chosen as simply the set of outcomes ; is the identity function; and

trivially. In effect, the random‐variable‐as‐measurable‐function approach of classical theory reduces to the “naive” or “realistic” method, in which the probabilities pertain to outcomes , and are not primarily inherited from some abstract measurable space .

Alternatively, let the sample space be and let be the class of Borel subsets of (so includes the singletons for each ). Define on by and

so is atomic. As before, with ,

Classical probability involves a quite heavy burden of sophisticated and complicated measure theory. There are good historical reasons for this, and it is unwise to gloss over it. In practice, however, the sample space , in which probability measure is specified, is often chosen—as above—in such a way that measure‐theoretic abstractions and complexities melt away, so that the “natural” or untutored approach, involving just outcomes and their probabilities, is applicable.

[MTRV] shows how to formulate an effective theory of probability which follows naturally from the naive or realistic approach described above, and which does not require the theory of measure as its foundation. The following pages are intended to convey the basic ideas of this approach.

Before moving on to this, here is an elaboration of a technical point of a financial character, which appeared in Example 5 above and in the ensuing discussion, and which is relevant in stochastic integration.