Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
Stochastic Processes and
Advanced Mathematical Finance
General Binomial Trees
Mathematicians Only: prolonged scenes of intense rigor.
for every . This says that the expectation, knowing the information up to time with respect to , unpicks in just the right way.
The purpose of this section is to set up a general formulation for understanding binomial processes. Ultimately the general formulation provides intuition for ﬁltrations, Radon-Nikodym derivatives, and Brownian motion
Consider the multi-step binomial process with probability on each branch represented in Figure 1. For comparison, also consider the multi-step binomial process with probability on each branch representing heads and probability on each branch representing tails in Figure 2.
Instead of branch probabilities, it is possible to assign the probability of each node in the tree. This is represented in Figure 3. This assignment gives the path probability measure instead. Given the branch probabilities, the path probabilities can be easily constructed. Given the path probabilities the branch probabilities can be recovered (if the probabilities are not or .)
For stochastic processes, a ﬁltration represents information about the process available up to and including each time through the sets of sample paths that satisfy all measurable conditions up to time . A ﬁltration has more information as time increases. That is, the set of measurable events for the stochastic process stays the same or increases as more conditions from the evolution of the stochastic processes become available.
Deﬁnition. A ﬁltration is an increasing sequence of -algebras on a measurable space . That is, given a measurable space (), a ﬁltration is a sequence of -algebras with for each and .
A stochastic process is said to be adapted to the ﬁltration if each is an integrable random variable that is measurable with the respect to the corresponding -algebra .
Example. For the multi-stage binomial process represented in Figure 3 the sample space, denoted
is the set of all possible inﬁnite sequences of s and s representing all possible outcomes of the composite experiment.
The -algebra denoted is the set of all possible sequences of 0’s and 1’s representing all possible outcomes of the composite experiment. We denote an element of as , where each or . That is, .
A ﬁltration adapted to the multi-stage binomial process is
In general is a set of sets. This ﬁltration represents the sets of sample paths that satisfy all measurable conditions in the binomial process up to time .
Next put a probability measure on the ﬁltration as the product of the probabilities along each branch of the binomial tree. So for example at the third level of the tree, the probabilities would be
The probabilities on paths, contain all the information about the probabilities along each path. For example at the third level, knowing the probabilities and the probabilities is suﬃcient to ﬁnd the branch probabilities , so long as none of are or .
Now suppose we had a diﬀerent measure on the paths with branch probabilities summarized in path probabilities and so on. As in the previous case, it is suﬃcient to specify the path probabilities, since the branch probabilities can be determined from that path information, so long as each are not or .
The likelihood ratios provide a way of mapping between the two measures and . Write the likelihood ratio as , called the Radon-Nikodym derivative of with respect to . Given the probability measure on paths, and the Radon-Nikodym derivative , the probability measure is the product . Note also that since the Radon-Nikodym derivative is deﬁned on paths, it is a random variable on the space . Even more, the Radon-Nikodym derivative is -adapted.
The case when some branch probability is or needs brief consideration. For example, if then and are both and no further information about paths through or is recoverable. Since all paths past the node have probability , that is to say, are impossible, the probabilities and are irrelevant anyway, so this is not a serious loss of information. If the probability measure is provided only for paths along which then we can recover the non-zero information about the corresponding branch probabilities.
Considering the probability measures and and the Radon-Nikodym derivative in the case when some is leads to an important deﬁnition. Then some path probabilities are , that is those paths are impossible. Suppose for the moment that with respect to the probability those same paths do not have probability . Then the Radon-Nikodym derivative is not deﬁned along those paths, and it is not possible to recover from . Restricting to the non-zero probability paths with respect to is not adequate, because that still loses information about the paths that are -impossible but are -possible.
This leads to a set of general deﬁnitions about probability measures.
Deﬁnition. If we say is absolutely continuous with respect to and write .
Note that an equivalent deﬁnition of absolute continuity is that .
Deﬁnition. Two probability measures on the space with -algebra are equivalent if for any set , if and only if . This is, the probability measures are equivalent if and .
Deﬁnition. If there is a with and we say is completely singular with respect to . Note that and so is completely singular with respect to and we write
and furthermore for any -measurable function
Note that the Radon-Nikodym derivative is deﬁned on paths, and is -measurable, so it is also a random variable. For simplicity, let . Then
for every . This says that the expectation, knowing the information up to time , with respect to accumulates in just the right way. The process represents the amount of change of measure so far up to time along the current path. If we want to know , it would be . If we want to know , then we would need the amount of change from time to time . That is just , which is change up to time with the change up to time removed. In other words
The point of the example is to apply the deﬁnitions and notations to the pair of binomial trees with deﬁned by and and and . This has minimal mathematical content, but is a good illustration of deﬁnitions and notation.
Calculating conditional probabilities with the simple deﬁnition from elementary probability remains easy on a point-by-point basis.
The Radon-Nikodym derivative along this tree is
This section is adapted from: “Chapter 2, Discrete Processes” in Financial Calculus by M. Baxter, A. Rennie, Cambridge University Press, Cambridge, 1996, .
holds for , , and .
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