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
General Binomial Trees
Rating
Mathematicians Only: prolonged scenes of intense rigor.
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Section Starter Question
Key Concepts
1. Given the probability measure P on paths, and the Radon-Nikodym
, the probability measure Q is the product dQ
P.
derivative dQ
dP
dP
2. Note that the Radon-Nikodym derivative is defined on paths, and is
F-measurable, so it is also a random variable.
3. Let ζt =
dQ
.
dP
Then
ζt = EP
dQ
| Ft
dP
for every t. This says that the expectation, knowing the information
in just the right way.
up to time t with respect to P, unpicks dQ
dP
4. If we want to know EQ [F (Xt ) | Fs ], then we would need the amount of
change from time s to time t. That is just ζt /ζs , which is change up to
time t with the change up to time s removed. In other words
EQ [F (Xt ) | Fs ] = ζs−1 EP [ζt F (Xt ) | Fs ] .
Vocabulary
1. A filtration is an increasing sequence of σ-algebras on a measurable
space.
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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 Ft .
3. The likelihood ratio as dQ
, called the Radon-Nikodym derivative
dP
of Q with respect to P. Given the probability measure P on paths, and
, the probability measure Q is the
the Radon-Nikodym derivative dQ
dP
dQ
product dP P.
4. If P [B] = 0 =⇒ Q [B] = 0 we say Q is absolutely continuous with
respect to P and write Q P.
5. Two probability measures on the space Ω with σ-algebra F are equivalent if for any set B ∈ F, P [B] > 0 if and only if QrobB > 0. This
is, the probability measures are equivalent if P Q and Q P.
6. If there is a B with P [B] = 0 and Q[B] = 1 we say Q
is completely
singular with respect to P. Note: P B C = 1 and Q B C = 0 so then
P is completely singular with respect to Q.
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 1/2 on each
branch represented in Figure 1. For comparison, also consider the multi-step
binomial process with probability 2/3 on each branch representing heads and
probability 1/3 on each branch representing tails in Figure 2.
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HHH
1/2
HH
1/2
HHT
1/2
H
1/2
1/2
HT
1/2
HTH
1/2
HTT
1/2
1/2
TH
1/2
THH
1/2
T
THT
1/2
1/2
TTH
TT
1/2
TTT
Figure 1: A binomial tree with probability 1/2 on each branch.
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HHH
2/3
HH
1/3
HHT
2/3
H
2/3
1/3
HT
2/3
HTH
1/3
HTT
1/3
2/3
TH
2/3
THH
1/3
T
THT
1/3
2/3
TTH
TT
1/3
TTT
Figure 2: A skewed binomial tree with probabilities 2/3 and 1/3 on each
branch.
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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.)
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, F), a filtration is
a sequence of σ-algebras {Ft }t≥0 with Ft ⊆ F for each t and t1 ≤ t2 =⇒
Ft1 ⊆ Ft2 .
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 Ft .
Example. For the multi-stage binomial process represented in Figure 3 the
sample space, denoted
Ω = {ω = (ωn )∞
n=1 : ωn = 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
F0
F1
F2
F3
= {∅, Ω}
= {∅, (0), (1), Ω}
= {∅, (0), (1), (00), (01), (10), (11), Ω}
= {∅, (0), (1), (00), (01), (10), (11), (000), (001), (010), (011), (100), (101), (110), (111), Ω}
In general Fn 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.
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π HHH
p31
π HH
p30
π HHT
p21
πH
p31
p20
π HT
p11
π HTH
p30
π HTT
p10
p31
p21
π TH
πT
π THH
p30
π THT
p20
p31
π TTH
π TT
p30
π TTT
Figure 3: A general
7 binomial tree.
Next put a probability measure P 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) p30 p20 p10 π000
(001) p30 p20 p11 π001
(010) p30 p21 p10 π010
(011) p30 p21 p11 π011
(100) p31 p20 p10 π100
(101) p31 p20 p11 π101
(110) p31 p21 p10 π110
(111) p31 p21 p11 π111
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.
Now suppose we had a different measure Q 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 φijk... /πijk... provide a way of mapping between the
, called the Radontwo measures P and Q. Write the likelihood ratio as dQ
dP
Nikodym derivative of Q with respect to P. Given the probability measure
P on paths, and the Radon-Nikodym derivative dQ
, the probability measure
dP
dQ
Q is the product dP P [·]. Note also that since the Radon-Nikodym derivative
dQ
is defined on paths, it is a random variable on the space Ω. Even more,
dP
the Radon-Nikodym derivative is Ft -adapted.
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 πijk... 6= 0 then we can recover the non-zero information about the
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φ HHH
φ HH
φ HHT
φ HTH
φH
φ HT
φ HTT
0
φ THH
φ TH
φT
φ THT
φ TTH
φ TT
φ TTT
Figure 4: Another general
binomial tree.
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φ HHH⁄π HHH
φ HH⁄π HH
φ HHT ⁄π HHT
φ HTH ⁄π HTH
φ H ⁄π H
φ HT ⁄π HT
φ HTT ⁄π HTT
φ THH ⁄π THH
φ TH⁄π TH
φ T ⁄π T
φ THT ⁄π THT
φ TTH ⁄π TTH
φ TT ⁄π TT
φ TTT ⁄π TTT
Figure 5: A general binomial tree 10
with the Radon-Nikodym derivative.
corresponding branch probabilities.
Considering the probability measures P and Q and the Radon-Nikodym
in the case when some pij is 0 leads to an important definition.
derivative dQ
dP
Then some path probabilities πijk... are 0, that is those paths are impossible.
Suppose for the moment that with respect to the probability Q those same
paths do not have probability 0. Then the Radon-Nikodym derivative dQ
dP
is not defined along those paths, and it is not possible to recover Q from
P. Restricting to the non-zero probability paths with respect to P is not
adequate, because that still loses Q information about the paths that are
P-impossible but are Q-possible.
This leads to a set of general definitions about probability measures.
Definition. If P [B] = 0 =⇒ Q [B] = 0 we say Q is absolutely continuous
with respect to P and write Q P.
Note that an equivalent definition of absolute continuity is that Q [B] >
0 =⇒ P [B] > 0.
Definition. Two probability measures on the space Ω with σ-algebra F are
equivalent if for any set B ∈ F, P [B] > 0 if and only if QrobB > 0. This
is, the probability measures are equivalent if P Q and Q P.
Definition. If there is a B with P [B] = 0 and Q [B]
1 we say Q isCcom C=
pletely singular with respect to P. Note that P B = 1 and Q B = 0
so P is completely singular with respect to Q and we write
P⊥Q
If
• P and Q are probability measures on common σ-algebra F; and
• Q is absolutely continuous with respect to P then there is a function
dQ
.
dP
and furthermore for any F-measurable function F
dQ
EQ [F (X)] = EP [F (X)L(X)] = EP F (X)
.
dP
L(x) =
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Note that the Radon-Nikodym derivative is defined on paths, and is F.
measurable, so it is also a random variable. For simplicity, let ζt = dQ
dP
Then
dQ
ζt = EP
| Ft
dP
for every t. This says that the expectation, knowing the information up to
time t, with respect to P accumulates dQ
in just the right way. The process
dP
ζt represents the amount of change of measure so far up to time t along the
current path. If we want to know EQ [F (Xt )], it would be EP [ζt F (Xt )]. If
we want to know EQ [F (Xt ) | Fs ], then we would need the amount of change
from time s to time t. That is just ζt /ζs , which is change up to time t with
the change up to time s removed. In other words
EQ [F (Xt ) | Fs ] = ζs−1 EP [ζt F (Xt ) | Fs ] .
(1)
Example. The point of the example is to apply the definitions and notations
to the pair of binomial trees with P defined by P [H] = 21 and P [T ] = 12 and
Q [H] = 23 and Q [T ] = 31 . 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.
EQ 1[X3≥1 ] | F2 X=0 = Q [X3 = 1 |HT, HT ]
= Q [HHH |HT, T H] =
=
Q [HHH]
Q [HT, T H]
2
8/27
= .
4/9
3
The Radon-Nikodym derivative along this tree is
t+X
t−X
t
t
2
2
4
2
ζt =
·
.
3
3
Now calculate the same conditional probabilities with formula (1) in Tables 1,
2,3.
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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Node
HH
HT
TH
TT
Expectation X2 Prob
2
· 1 + 31 · 1
2 1
3
2
1
·1+ 3 ·0
1 23
3
2
· 1 + 31 · 0
1
3
2
1
·0+ 3 ·0
0 0
3
Table 1: EQ 1[X3≥1 ] | F2
Node
HH
HT
TH
TT
Expectation
X2 Prob
4 3
4 2 2
1
4 2
1
·1· 3 + 2 ·1· 3
2
2
3
3
2 2 1
2
2
1
· 1 · 34
+ 2 · 0 · 43 32
1 12 34
2
3
2 2 1
2
2
1
· 1 · 34
+ 2 · 0 · 43 32
1 12 34
2
3
2
3
1
· 0 · 34 32 + 12 · 0 · 23
0 0
2
Table 2: EP 1[X3 ≥1] ζ3 | F2
X2 EQ 1[X3≥1 ] |F2 EP 1[X3 ≥1] ζ3 |F2
4 2
2 1
3
1
4 2 2
1 32
2
3
3
0 0
0
ζ2
4 2
3
4
2
3 3
2 2
3
2
3
2
3
Prob
1
2
3
0
Table 3: Summary of EQ [F (Xt ) | Fs ] = ζs−1 EP [ζt F (Xt ) | Fs ].
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Problems to Work for Understanding
1. On the general binomial tree, prove that
dP
ζt = EP
| Ft
dQ
holds for t = 1, 2, and 3.
2.
3.
4.
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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Outside Readings and Links:
1.
2.
3.
4.
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Steve Dunbar’s Home Page, http://www.math.unl.edu/~sdunbar1
Email to Steve Dunbar, sdunbar1 at unl dot edu
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