Partitions and information

For our next topic it is better to extend the time scale to three steps. This gives the appropriate sample space as

Ω = {uuu, uud, udu, udd, duu, dud, ddu, ddd}.

The probability will be defined as before, so

P(uuu)= p³,

P(uud)= P(udu) = P(duu) = p²(1− p), P(udd)= P(dud) = P(ddu) = p(1 − p)², P(ddd)= (1 − p)³.

We have to modify the definition of K₁, K2and add the third return K₃with value U for triples ending with u and D otherwise, so that Kndepends only on the nth element ofω, just as in the single-step binomial model. However, the notion of independence is now somewhat more involved, as clarified by the following exercise.

Exercise 3.2 Prove that K₁, K2, K3are independent, which by defini-tion means that for each pair we have condidefini-tion (3.1) and also

P ³

k=1

{ω : Kⁱ= xⁱ}

=

3 k=1

P({ω : Kⁱ= xⁱ}).

The third stock price takes the form

S (3)= S (2)(1 + K3)=

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪

⎪⎩

S^uuu= S (0)(1 + U)³,

S^uud= S^udu= S^duu= S (0)(1 + U)²(1+ D), S^ddu= S^dud= S^udd= S (0)(1 + U)(1 + D)², S^ddd= S (0)(1 + D)³.

Within the numerical scheme of the previous section we have the com-plete picture

S(0) S(1) S(2) S(3)

0.6 172.8

0.6 144

0.6 120

0.4

0.6 129.6 100

0.4

0.6 108

0.4

90

0.4

0.6 97.2

0.4

81

0.4

72.9 Consider the following random variable

H= (S (3) − 100)⁺

which is nothing but a call payoff with strike K = 100.

If we own such a security we might be interested in the expected payoff so we compute

E(H) = 72.8 × P({uuu}) + 29.6 × P({uud, udu, duu}) + 0 × P({udd, dud, ddu}) + 0 × P({ddd})

= 28.512.

Exercise 3.3 Extend the pricing scheme of the previous section to find the option price and observe that it is not equal to the expectation computed above.

Next we analyse the future possible developments and their impact on our views. It is important to emphasise that we are now performing a ‘what if’ analysis. We do not travel in time but consider hypothetically all pos-sible future turns of events with the purpose of revising our point of view and possibly responding with some appropriate action.

Position at step 1 Case ‘up’

Let us prepare ourselves for the case of an increase in the stock price after one step. This means that the collection of all scenarios available

will be reduced to the triples beginning with u, which we denote by B_u = {uuu, uud, udu, udd}. Knowledge that the first step is ‘up’ means that this set will play the role ofΩ and a probability is produced by adjusting the original probabilities to ensure that the new probability of B_u is 1. For ω ∈ Buwe put

P_u(ω) = P(ω) P(B_u).

Observe that u is the first element of eachω ∈ Bu and according to the definition of P, we have P(Bu) = p and this is the first factor producing P(ω), so it will cancel. For instance

P_u(udu)= P(udu)

P(B_u) = p(1− p)p

p = (1 − p)p.

Of course P_u(B_u)= _P(B¹_u)

ω∈BuP(ω) = 1. Observe that for A ⊆ Bu, P_u(A)= P(A∩ Bu)

P(B_u) = P(A|Bu)

so P_uis what is called, quite generally, the conditional probability P(·|Bu), considered for subsets of B_u.

Next we compute the expectation of H in this new probability space. We consider values of H corresponding to all four elements of Bu, and use a natural version of the, by now familiar, formula for the expectation:



ω∈Bu

H(ω)Pu(ω) = 72.8 × 0.6²+ 29.6 × 2 × 0.6 × 0.4 + 0 × 0.4²= 40.416 which would be good news for the option holder.

We introduce the following two alternative notations for this (condi-tional) expectation:

E(H|Bu)= E(H|S (1) = 120) = 40.416.

Case ‘down’

In the case of a down move at the first step we introduce, similarly, the set B_dof all still remaining scenarios (all those beginning with a d) and the adjusted probabilities forω ∈ Bd:

P_d(ω) = P(ω) P(B_d). Clearly, the expectation of H in this situation is

E(H|Bd)= E(H|S (1) = 90) = 29.6 × p²= 10.656, a bit disappointing for anyone who owns such a security

The two cases considered decompose all scenarios into two groups.

Mathematically, we have a so-called partition of Ω, meaning that Ω = B_u∪ Bd, and the components are disjoint, which motivates the following general definition.

Definition 3.5

A familyP = {Bi} of subsets of Ω is a partition of Ω if Bi∩ Bj = ∅ for i j and Ω =

Bi.

The partition defined at the first step will be denoted P1= {Bu, Bd}.

We put together these two cases defining a random variable with two com-peting notations, equivalent since the partitionP1 is fully determined by S (1),

E(H|P1)(ω) = E(H|S (1))(ω) =

 40.416 if ω ∈ Bu, 10.656 if ω ∈ Bd. Position at step 2

Consider all possible price movements in the first two steps. There are four cases, which can be described by specifying a partition

P2= {Buu, Bud, Bdu, Bdd} ofΩ into four disjoint sets of paths:

B_uu= {uuu, uud}, B_ud= {udu, udd}, B_du= {duu, dud}, B_dd= {ddu, ddd}.

Each of these is equipped with probabilities defined as before. Observe the effect of the cancellation mentioned above and notice for example that P(B_ud)= p(1 − p) so Pud(udu)= p, Pud(udd)= 1 − p and in each case we have a well-known single-step binomial tree.

As before, we compute the expected value of H in each case. Clearly E(H|Buu)= 55.52,

E(H|Bud)= 17.76, E(H|Bdu)= 17.76, E(H|Bdd)= 0,

hence

E(H|P2)(ω) =⎧⎪⎪⎪⎨

⎪⎪⎪⎩

55.52 if ω ∈ Buu, 17.76 if ω ∈ Bud∪ Bdu, 0 ifω ∈ Bdd.

Remark 3.6

We can see that for the middle prices we get the same value on each of the two corresponding sets and so we could employ the alternative, more intuitive notationE(H|S (2)). To this end, note that the partition generated by the values of S (2) is different from the partition related to the history of the movements over the first two steps, introduced above. The partition generated by S (2) consists of just three elements: B_uu, Bud∪ Bdu, Bdd, and the random variableE(H|S (2)) takes three values. The fact that the values E(H|Bud), E(H|Bdu) coincide gives the equivalence of these two approaches in this instance, but this does not have to be the case in general.

The actual values of S (2) are irrelevant for defining the conditional ex-pectation since they do not appear in the computations. What matters is the partition related to these values, which just play the discerning or di fferen-tiating role.

Position at step 3

Given the knowledge of all three steps we will know the actual payoff.

The above analysis can formally be performed for the partition P3= {Buuu, . . . , Bddd}

ofΩ into single-element parts with no randomness and obvious conclusion that

E(H|P3)= H.

We are in a position to give a general definition.

Definition 3.7

For any random variable X and any event B such that P(B) 0 the condi-tional expectation of X given B is defined by

E(X|B) =

ω∈Ω

X(ω)P(ω|B)

where P(ω|B) = _P(B)¹ P(ω) for ω ∈ B and P(ω|B) = 0 otherwise. Given a partitionP = {Bi}, the above definition applied to each Bigives a function defined for eachω in Ω, constant on each Bi,

E(X|P): Ω → R

by assigning

Bi ω → E(X|Bi).

This random variable is called the conditional expectation of X with respect to the partition{Bi}.

Properties ofE(X|P) are discussed in a general setting in the next chap-ter. Here we continue the analysis of binomial trees.