Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Stochastic Processes and
Advanced Mathematical Finance

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General Binomial Trees

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Rating

Rating

Mathematicians Only: prolonged scenes of intense rigor.

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Section Starter Question

Section Starter Question

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Key Concepts

Key Concepts

  1. Given the probability measure on paths, and the Radon-Nikodym derivative d d , the probability measure is the product d d .
  2. Note that the Radon-Nikodym derivative is defined on paths, and is -measurable, so it is also a random variable.
  3. Let ζt = d d . Then
    ζt = 𝔼 d d |t

    for every t. This says that the expectation, knowing the information up to time t with respect to , unpicks d d in just the right way.

  4. If we want to know 𝔼 F(Xt)|s, then we would need the amount of change from time s to time t. That is just ζtζs, which is change up to time t with the change up to time s removed. In other words
    𝔼 F(Xt)|s = ζs1𝔼 ζtF(Xt)|s .

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Vocabulary

Vocabulary

  1. A filtration is an increasing sequence of σ-algebras on a measurable space.
  2. A stochastic process Xt is said to be adapted to the filtration if each Xt is an integrable random variable that is measurable with respect to the corresponding σ-algebra t.
  3. The likelihood ratio as d d , called the Radon-Nikodym derivative of with respect to . Given the probability measure on paths, and the Radon-Nikodym derivative d d , the probability measure is the product d d .
  4. If B = 0 B = 0 we say is absolutely continuous with respect to and write .
  5. Two probability measures on the space Ω with σ-algebra are equivalent if for any set B , B > 0 if and only if QrobB > 0. This is, the probability measures are equivalent if and .
  6. If there is a B with B = 0 and B = 1 we say is completely singular with respect to . Note: BC = 1 and BC = 0 so then is completely singular with respect to .

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Mathematical Ideas

Mathematical Ideas

General Binomial Trees

Path Probabilities and Filtrations

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 12 on each branch represented in Figure 1. For comparison, also consider the multi-step binomial process with probability 23 on each branch representing heads and probability 13 on each branch representing tails in Figure 2.


General binomial tree with baranches 1/2

Figure 1: A binomial tree with probability 12 on each branch.


General binomial tree with branches 2/3

Figure 2: A skewed binomial tree with probabilities 23 and 13 on each branch.

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 0 or 1.)


General binomial tree

Figure 3: A general binomial tree.

For stochastic processes, a filtration represents information about the process available up to and including each time t through the sets of sample paths that satisfy all measurable conditions up to time t. 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 σ-algebras on a measurable space Ω. That is, given a measurable space (Ω,), a filtration is a sequence of σ-algebras {t}t0 with t for each t and t1 t2t1 t2.

A stochastic process Xt is said to be adapted to the filtration if each Xt is an integrable random variable that is measurable with the respect to the corresponding σ-algebra t.

Example. For the multi-stage binomial process represented in Figure 3 the sample space, denoted

Ω = ω = (ωn)n=1 : ω n = 0, 1 for all n.

is the set of all possible infinite sequences of 0s and 1s representing all possible outcomes of the composite experiment.

The σ-algebra denoted Ωn is the set of all possible sequences of n 0’s and 1’s representing all possible outcomes of the composite experiment. We denote an element of Ωn as ω = (ω1,,ωn), where each ωk = 0 or 1. That is, Ωn = 0, 1 n.

A filtration adapted to the multi-stage binomial process is

0 = {, Ω} 1 = {, (0), (1), Ω} 2 = {, (0), (1), (00), (01), (10), (11), Ω} 3 = {, (0), (1), (00), (01), (10), (11), (000), (001), (010), (011), (100), (101), (110), (111), Ω}

In general n is a set of 2n+1 sets. This filtration represents the sets of sample paths that satisfy all measurable conditions in the binomial process up to time t.

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

(000)p30p20p10π000
(001) p30p20p11π001
(010) p30p21p10π010
(011) p30p21p11π011
(100) p31p20p10π100
(101) p31p20p11π101
(110) p31p21p10π110
(111) p31p21p11π111

The probabilities on paths, π0,π1,π10,π11,π01,,π000,, contain all the information about the probabilities along each path. For example at the third level, knowing the 8 probabilities π000,,π111 and the 4 probabilities π11,π10,π01,π00 is sufficient to find the 8 branch probabilities p10,p11,,p31, so long as none of π00,,π11 are 0 or 1.


Another general binomial tree

Figure 4: Another general binomial tree.

Now suppose we had a different measure on the paths with branch probabilities qij summarized in path probabilities ϕijk 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 ϕijk are not 0 or 1.

The likelihood ratios ϕijkπijkprovide a way of mapping between the two measures and . Write the likelihood ratio as d d , called the Radon-Nikodym derivative of with respect to . Given the probability measure on paths, and the Radon-Nikodym derivative d d , the probability measure is the product d d . Note also that since the Radon-Nikodym derivative d d is defined on paths, it is a random variable on the space Ω. Even more, the Radon-Nikodym derivative is t-adapted.


Radon-Nikodym derivaive

Figure 5: A general binomial tree with the Radon-Nikodym derivative.

Two Probability Measures

The case when some branch probability pij is 0 or 1 needs brief consideration. For example, if p21 = 0 then π111 and π110 are both 0 and no further information about paths through π111 or π110 is recoverable. Since all paths past the node (11) have probability 0, that is to say, are impossible, the probabilities p21 and p20 are irrelevant anyway, so this is not a serious loss of information. If the probability measure is provided only for paths along which πijk0 then we can recover the non-zero information about the corresponding branch probabilities.

Considering the probability measures and and the Radon-Nikodym derivative d d in the case when some pij is 0 leads to an important definition. Then some path probabilities πijk are 0, that is those paths are impossible. Suppose for the moment that with respect to the probability those same paths do not have probability 0. Then the Radon-Nikodym derivative d d 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 B = 0 B = 0 we say is absolutely continuous with respect to and write .

Note that an equivalent definition of absolute continuity is that B > 0 B > 0.

Definition. Two probability measures on the space Ω with σ-algebra are equivalent if for any set B , B > 0 if and only if QrobB > 0. This is, the probability measures are equivalent if and .

Definition. If there is a B with B = 0 and B = 1 we say is completely singular with respect to . Note that BC = 1 and BC = 0 so is completely singular with respect to and we write

If

L(x) = d d .

and furthermore for any -measurable function F

𝔼 F(X) = 𝔼 F(X)L(X) = 𝔼 F(X)d d .

Note that the Radon-Nikodym derivative is defined on paths, and is -measurable, so it is also a random variable. For simplicity, let ζt = d d . Then

ζt = 𝔼 d d |

for every t. This says that the expectation, knowing the information up to time t, with respect to accumulates d d in just the right way. The process ζt represents the amount of change of measure so far up to time t along the current path. If we want to know 𝔼 F(Xt), it would be 𝔼 ζtF(Xt). If we want to know 𝔼 F(Xt)|s, then we would need the amount of change from time s to time t. That is just ζtζs, which is change up to time t with the change up to time s removed. In other words

𝔼 F(Xt)|s = ζs1𝔼 ζtF(Xt)|s . (1)

Example.

The point of the example is to apply the definitions and notations to the pair of binomial trees with defined by H = 1 2 and T = 1 2 and H = 2 3 and T = 1 3. 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.

𝔼 1[X31]|2 X=0 = X3 = 1|HT,HT = HHH|HT,TH = HHH HT,TH = 827 49 = 2 3.

The Radon-Nikodym derivative along this tree is

ζt = 4 3 t+Xt 2 2 3 tXt 2 .

Now calculate the same conditional probabilities with formula (1) in Tables 1, 2,3.


NodeExpectation X2Prob




HH 2 3 1 + 1 3 1 21
HT 2 3 1 + 1 3 0 12 3
TH 2 3 1 + 1 3 0 1
TT 2 3 0 + 1 3 0 00

Table 1: 𝔼 1[X31]|2


NodeExpectation X2Prob




HH 1 2 1 4 3 3 + 1 2 1 4 3 2 2 32 4 3 2
HT 1 2 1 4 3 2 2 3 + 1 2 0 4 3 2 3 2 11 2 4 3 22 3
TH 1 2 1 4 3 2 2 3 + 1 2 0 4 3 2 3 2 11 2 4 3 22 3
TT 1 2 0 4 3 2 3 2 + 1 2 0 2 3 3 00

Table 2: 𝔼 1[X31]ζ3|2


X2𝔼 1[X31]|2𝔼 1[X31]ζ3|2ζ2 Prob





21 4 3 2 4 3 2 1
12 3 1 2 4 3 22 3 4 3 2 32 3
000 2 3 2 0

Table 3: Summary of 𝔼 F(Xt)|s = ζs1𝔼 ζ tF(Xt)|s.

Sources

This section is adapted from: “Chapter 2, Discrete Processes” in Financial Calculus by M. Baxter, A. Rennie, Cambridge University Press, Cambridge, 1996, [1].

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Problems to Work

Problems to Work for Understanding

  1. On the general binomial tree, prove that
    ζt = 𝔼 d d|

    holds for t = 1, 2, and 3.

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Books

Reading Suggestion:

References

[1]   M. Baxter and A. Rennie. Financial Calculus: An introduction to derivative pricing. Cambridge University Press, 1996. HG 6024 A2W554.

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Links

Outside Readings and Links:

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Last modified: Processed from LATEX source on February 11, 2016