Comment on: "Computation of deterministic volatility surfaces", by N. Jackson, E. Süli, and S. Howison, Vol. 2(2) (Winter, 1998/99), pp. 5-32

Mark Rubinstein

Abstract

On page 7 of this paper appears the following statement:

"Algorithm [16] assumes that volatility only has a skew structure, i.e. that ó = ó (S)."

Reference [16] is to my paper "Implied binomial trees" (Journal of Finance, July 1994). Since this statement is not only false but also suggests that my approach is much narrower than it is, I felt compelled to write this reply.

In fact, the model in my paper is general enough to allow ó = ó (S, t) - indeed it must allow this dependence on time for a solution to exist. In several places in my paper, most obviously Table V, pages 798-799, and the related discussion, it is shown explicitly that the local volatility depends on both the concurrent underlying asset price and time. This table is a matrix of local volatilities with asset price down the left and time along the top. What could be clearer!

I can only surmise that the authors of the above paper did not trouble to read my paper but concluded that ó = ó (S) because they know that my implied tree is based on options maturing at a single date. However their conclusion does not follow from this fact.1 Indeed one of the rich implications of the model is that, under its assumptions, both the asset price dependence and term structure of local volatility can be inferred just from the prices of European options on the same underlying asset, expiring on the same date, but with differences in strike prices.

1 Admittedly, this is a limitation. However, compared to other approaches, it buys the model the assurance that the interior structure of the tree cannot suffer from negative probabilities and permits a very simple recursive procedure, which the above authors also fail to mention in their model comparison discussion.