fin-book-09 - Mathematics of Finance Course Notes Donald G....

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Unformatted text preview: Mathematics of Finance Course Notes Donald G. Saari ii c 2008 by Donald G. Saari Departments of Economics and of Mathematics University of California at Irvine Irvine, California 92697 dsaari@uci.edu Introduction A particular delight about this area of the Mathematics of Finance is that while much is known, it also is true that not very much is known. Instead, the current state of affairs almost demands treating all of the existing models that will be discussed here as first, crude attempts to provide some structure, some sense, some science to that vast amount of data and uncertainty which characterizes financial interactions. This reality that the current status of the field relies upon rough models, models that most certainly will be viewed in the future as first attempts, forms an argument that it is crucial for us to learn to be skeptical. Namely, dont fully accept anything. In any area characterized by these traits, this lack of a mature under- standing of the field, provides excitement which is generated by the inherient dangers and the hidden opportunities. The dangers arise from the reality that serious errors can be, and are, made. Indeed, errors can and must be expected whenever existing models, or the mathematics of what is known is applied in inappropriate settings. In this area of finance, an error can have the pragmatic effect of translating into a considerable loss of money. The opportunities exist because so much is unknown and is waiting to be discovered. But to exploit these opportunities, a first step is to understand how and why the existing models are limited. In a practical sense, this requires knowing when a particular model may provide wrong answers so something else should be used. In a theoretical sense, this means that models need to be corrected, extended, or changed to allow them to be legitimately applied to new settings. What is required to handle such a topic and the novel situations they present? First and foremost, it is to develop the habit of critical thinking. This means that rather than accepting given assumptions rather than following the traditional approach expected in most economics and math- ematics courses where a student fully accepts whatever is said and then concentrates on details all assumptions should be carefully and critically examined. The attitude should be, Do you really believe them? Do they iii iv INTRODUCTION make sense? When and where might they be valid? When and where might they be wrong ? What impact do each of the individual assumptions have on the conclusions? Would different assumptions create conclusions which are more in keeping with what we observe? Whenever the assumptions are varied, now what happens?...
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fin-book-09 - Mathematics of Finance Course Notes Donald G....

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