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Options and Option Values Under Incomplete Information
October 3, 2001
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Abstract:
Premise: We will assume that at any time there is a well defined predictive distribution of probabilities attaching to realizations, but that this distribution is not known. While we must and will assume that we know something in advance, we will not assume that it is enough to specify the predictive distribution conditional on the current state of the world. What we want to assume is, in some sense, that we have somewhat fuzzy and ambiguous information that bears on, and sheds light on, the predictive distribution. Promise: This is not a mathematical problem, as we would usually understand that term. It is a management problem, and what we will talk about here are tools for using what is known, or believed to be known, in a way -- in a somewhat impressionistic but nonetheless quite functional way -- that minimizes the risk of placing too much faith in the model, while maximizing the usefulness of all prior information. Business background: Our business is to develop unconventional option models adapted to the needs of market professionals. The ideas we will talk about here come out of our research on financial markets and our product development work. These ideas have concrete versions that are option models we sell to the public. I mention this not as an advertisement, but by way of truth in public speaking. We expect and hope that some of you in the audience will be inspired to adopt this very flexible strategy for option modeling, and that it will be as productive in your hands as it has been for us. We would however be remiss if we did not extend an offer to work with you and with your employers and employers-to-be on a conventional fee basis. The ideas we will be presenting are not new or novel to us. In various forms they have been used for years. It is our hope that we can nonetheless advance the state of the art. Lastly, we want to provide a broad survey of this very extensive subject, and so we will not dwell too long on any single part. We leave it to the audience to stop us with questions that explore in more depth aspects of particular interest. We have also provided some readings that provide further insight into some parts of this broad subject. Approaches. We have in mind two quite different approaches:
The first topic alone will however keep us fully occupied this evening. We leave the subject of empirical models to a later talk. State Dependent Distributions. The idea is very simple: that the space in which outcomes are realized can be viewed as a sort of manifold of local distributions. Assuming for the moment that we are satisfied that we know what sort of distributions the local distributions are, there remains uncertainty about which locale we will be in. It is necessary in practice to assume that the state space is discrete. If we accept this model of the world, we see that there is a discrete state space with some sort of matrix of transition probabilities among them. It might be possible to tackle this as an empirical problem, and we will present some findings of that sort. It is also possible to import non-quantitative prior information into the model. While this sounds unscientific, it may in fact be inevitable, because the data demands to estimate the transition matrix tend to be rather massive. There is a second facet of models of this sort, and that is to use a more flexible family of locale distribution functions, and not just assume a linear diffusion. We have developed models of bounded diffusion which have the property that the "local" distribution actually stays local. We will also present some thoughts on how specific versions of state dependent models correspond to some interesting, intuitively appealing global models. Organization of This Talk
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