Overview of basic discrete probability approximation construction pro-

procedure

Creating discrete probability-based approximations for probability models is well known in the context of option pricing. Cox, Ross and Rubinstein (CRR) [21] pioneered the lattice approach. They developed a discrete time binomial approach to price options. Boyle [15] took the CRR methodology a step further and proposed a trinomial option pricing model. In [44] the discrete probability-approximated values are shown to converge to the true option values as number of time step divisions goes to infinity. Creating multinomial discrete probability approximation with positive probabilities for an arbitrary correlation structure is a hot topic of research. Reference [45] provides a fairly generalized methodology to modelk sources of uncertainty in the form of anumericaldiscrete probability approximation.

The most basic discrete probability approximation construction procedure revolves around the idea of moment matching. The thesis however, aims to construct fairly general discrete probability approximation which could use any possible statistical feature as in Chapter 5. It is very similar to how we construct lattices to price options.

LetΥrepresent a discrete probability approximation15_{for a probability space inℵdimen-}

sions. Then for any⊕ ∈Υwe have⊕= (⊕₁,⊕₂) where| ⊕₁|=ℵand⊕₂ ∈[0,1]. Especially, a pair in a discrete probability approximation is a joint evolution of risky growth vector coupled together with a probability where the individual probabilities sum to one.

Tree in 1-D

Ifℵ =1 the possible example ofΥare ((u,p),(d,1−p)),((u,p),(m,q),(l,1−p−q)) etc: many distinct discrete probability approximation are possible in one dimension.

The discrete probability approximation is constructed so that moments are matched. Con- sider ((u,p),(m,q),(l,1− p−q)) for example. This group has five unknowns and so we will need typically five equations to specify these unknowns16. In general, for aΥwithgunknowns (probability of branches and stochastic evolution of random variable ) requiresgequations and hence the firstgmoments must be matched.

15_{The thesis is interested in discrete probability approximation for risky growth vector over an interval∆T.} 16_{Generally forzunknowns we requireznon-linear equations to have a finite dimensional solution set which is}

3.8. On discrete probability approximations 39 d u Approximation f rT 1-p p

Figure 3.9: A discrete probability approximation approximation in 1-D. Here rT say risky

growth over an interval is variable being approximated.

Figure 3.9 shows a continuous probability model of risky growth approximated by a dis- crete probability approximation. The discrete probability approximation is ((u,p),(d,1− p)). As such three parameters must be found. These could be found by matching the first three moments as below: up+d(1− p) = E(rT) (3.58) u2p+d2(1− p) = E(r_T2) (3.59) u3p+d3(1− p) = E(r_T3) (3.60) Tree in 2-D Ifℵ =2 a possible example ofΥis (((u1,u2),p),((u1,d2),q),((d1,u2),r),((d1,d2),1− p−q−r)).

Note that theΥis a discrete probability approximation for joint evolution of risky growths and it implies discrete probability approximation of one dimension lower for the respective risky growths. The idea is illustrated in Figure 3.10 where a parent discrete probability approxima- tion branches offinto two sub-trees. The discrete probability approximation for risky growth of asset one would be ((u1,p+q),(d1,1− p−q)). The discrete probability approximation for

risky growth of asset two would be ((u2,p+r),(d2,1− p−r)).

The discrete probability approximation Υ has seven unknowns in the above example. To fill in the values we will need seven equations. It is simple to get seven equations by matching first 3 moments of discrete probability approximation of lower dimension and then usingΥto match the cross-moment. Figure 3.10 illustrates the ideas discussed.

40Chapter3. Introduction to discrete probability approximation and sketch of modeling approach + (u1,u2) (u1,d2) (d1,u2) (d1,d2) u1 d1 u2 d2 p q r 1-p-q-r p+q 1-p-q p+r 1-p-r

PARENT TREE RESULTANT SUB-TREES

Figure 3.10: Illustrating discrete probability approximation construction in 2-D for correlated variables.

Tree in 3-D

If ℵ = 3 it is obvious that discrete probability approximation at the highest level has eight branches for up/down branching for a lowest level discrete probability approximation, imply- ing three trees with four branches at a lower level. This further implies three trees with two branches at the lowest level. If our original tree had say thirteen unknowns then we need thir- teen equations. Since 13=9+3+1 we could get all the equation by matching first three moments for the three trees at the lowest level, cross moments for three trees at a higher level and cross moment for one discrete probability approximation at the highest level. Figure 3.11 illustrates the ideas discussed with a parent tree implying sub-trees. There are many possible ways to construct trees in 3-D if lower level trees are allowed to have more arbitrary number of states. For instance, instead of just up/down we might have up/middle/down. If the dynamics of the asset prices are dependent on state variables we could easily accommodate this by having trees dependent on state variables.

General framework for a discrete probability approximation inℵ-D

Binomial lowest level discrete probability approximation

The most general form for Υat the highest level would have 2ℵbranches if only up/down movements are allowed for lowest level discrete probability approximation. With ℵ

2

!

discrete probability approximation at a lower level with four branches to match cross moments. At the lowest level we haveℵdiscrete probability approximations with just 2 branches.

To fill outΥwe need 2ℵ₋₁+_{2ℵequations which could be obtained by matching moments}

from trees at different levels. This is because 2ℵ− 1 parameters for probabilities and 2ℵ for states.

3.8. On discrete probability approximations 41 + (u1,u2,u3) (u1,u2,d3) (u1,d2,u3) (u1,d2,d3) (d1,u2,u3) (d1,u2,d3) (d1,d2,u3) (d1,d2,d3) (u1,u2) (u1,d2) (d1,u2) (d1,d2) u1 d1 p q r s t u v 1-p-q-r-s-t-u-v p+q r+s t+u 1-p-q-r-s-t-u p+q+r+s 1-p-q-r-s

PARENT TREE SUB-TREES SUB-SUB-TREES

+

Figure 3.11: Illustrating discrete probability approximation construction in 3-D for correlated variables.

ℵ+ℵ(ℵ−1)= ℵ2_{third moments. Asℵincreases at first three moments are exactly enough, then}

more than enough, but eventually not nearly enough to capture all the probability outcomes. If we are aiming for five or fewer dimensions then three moments are enough.

If more than up/down movements are allowed for lowest level trees we will have many more equations!

Trinomial lowest level trees

This analysis follows a discussion similar to that given for the binomial case. To fill out Υ we need 3ℵ−1+3ℵ equations which could be obtained by matching moments from trees at different levels.This is because 3ℵ_{−1 parameters for probabilities and 3ℵfor states.}

The above discussion provided a general framework. Note it is possible to create trees with nice structure at a cost of low moment matching. We could do this by pre-filling the probabilities. Lets give an example using a correlated set of standard normal random variables

Ziin 3-D. Suppose the standard normals could either take a value of -1 or+1. Also lets denote

ρi j as the correlation between a pair of standardd normals i and j. A very simple discrete

probability approximation structure is given in Table 3.2.

As we see there is a lot of innovation in which we could match cross moments. Are we just interested in matching the entries of a covariance matrix? or are interesting in matching some creative cross-moments? All just depends upon the degree of accuracy we seek to achieve. For some problems, for instance, second order moment matching might suffice. Consider a single period Markovitz objective. It explicitly contains just the mean and variance, so any discrepancies in the higher moments is not relevant. In a more formal notation consider two risky assets who have the same expected return over a period that is:

42Chapter3. Introduction to discrete probability approximation and sketch of modeling approach Outcome Probability uuu 1+ρ12+ρ23+ρ31 8 uud 1+ρ12−ρ23−ρ31 8 udu 1−ρ12−ρ23+ρ31 8 udd 1−ρ12+ρ23−ρ31 8 duu 1−ρ12+ρ23−ρ31 8 dud 1−ρ12−ρ23+ρ31 8 ddu 1+ρ12−ρ23−ρ31 8 ddd 1+ρ12+ρ23+ρ31 8

Table 3.2: A nice discrete probability approximation structure with correlated standard normals in 3-D

E[r1,T]= R,E[r2,T]= R (3.61)

The investor allocates a fractionαso as to minimize the variability in total returns that is:

min

α Var[rT] = minα Var[αr1,T +(1−α)r2,T] (3.62)

= min

α (α

2_(E[r2

1,T]−E[r1,T]2)+(1−α)2(E[r₂2_,T]−E[r2,T]2)

+2α(1−α)(E[r1,Tr2,T]−E[r1,T]E[r2,T])) (3.63)

The above example could be generalized to an N dimensional singleMarkowitzportfolio problem and it can easily be seen that it is distribution independent and all what we require is that moments/cross moments up to order two must be matched. As we see for most practical purposes trees do provide a reasonable approximation.

If we let the time step division ∆T → 0 we approach a continuous time behavior under a suitable moment matching scheme. Forwell-behavedutility functions we could approach the continuous time solution implied by using exact distributions. We will use a binomial discrete probability approximation with moment matching of order two to illustrate such a behavior in Figure 3.12 and 3.13. The transaction cost structure is the standard Davis and Norman model (see [28]). We use CRRA and log-utilities. Convergence behavior is illustrated in Figures 3.12 and 3.13 for a suitable choice of parameters and utility functions.