## Info

[V(i + 1, j + l)Pup + V(i + 1, j - 1 )Pdown + V(i + l,j)pm]

1 + rAt where the coefficients pup, pm and pdown have been defined to be

Pdown

respectively. From a technical point of view one can notice that, for the chosen transformation, the coefficients pup, pm and pdown are independent of the state variable <f> and can therefore be calculated once and for all, rather than 'on the fly' during the backward induction process. But it is conceptually more important to notice that the choice of the symbols pup, pm and Pdown for these three coefficients, clearly suggestive of a probabilistic interpretation, was not without a certain degree of foresight, since one can immediately verify that it is always true that

If, furthermore, pap, pm and Pdown are all positive, then the interpretation of these coefficients as (risk-adjusted) 'probabilities' of moving to an up, middling or down state is. warranted. In addition, if one compares two binomial moves on a binomial lattice with a single trinomial move on a FD mesh, one can immediately notice the fundamental similarity between the two approaches (see Figure 9.3).

An approach like the one described in Chapter 8, and routinely implemented for models like BDT, is in fact tantamount to prescribing a priori the probabilities, and achieving the matching of the moments by moving the grid coordinates; a finite-differences scheme, on the other hand, pre-assigns these coordinates and strives to find the probabilities that ensure the matching of the moments.

Furthermore, one can prove (Ames (1977)) that sufficient condition for convergence of the EFI) scheme is that all the coefficients (i.e. 'probabilities') Pup,Pm and Piown should simultaneously be positive. The intuition behind this requirement is quite easy to grasp: Equation (9.11) implicitly enforces the matching of the first two moments (conditional expectation and variance) of the quantity V. Whenever the expectation lies outside the range [V(/+ 1,7 — 1) V(i + 1, j + 1)], no positive choice for all the 'probabilities' can produce

Figure 9.3 The similarity between two steps in a binomial tree, which reach from A the three points E, D and F, and a single backward move in the EFD method, which links E, D and F with A

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