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Dimensionality: Please Explain :)

I'm not sure how to understand/interpret the "dimensionality" of a pricing problem.

Is it just the number of stochastic variables or do the number of times the underlying is observed affect dimensionality?

e.g. a single-asset geometric brownian barrier option monitered discretely at the end of each day for 30 days - is that 30 dimensional or 1 dimensional?

e.g.2. a 2-asset spread option (geometric brownians) which only pays if the spread is above a certain level (barrier) for N days in a row (monitered at the end of each day) - is this 2 dimensional or 2T dimensional?

Really appreciate your time :)
Try harder. A brief google search will give you any number of articles to read through.


ahh but both links refer to the dimensionality of integrals without explaining which factors influence the dimensionality / the formulation of the integral in relation to time and stochastic factors. It's kinda like me asking you where milk comes from and you reply with a link to Walmart saying "try harder". Maybe it's my fault for not knowing the basics properly but i sincerely don't know whether each monitering of a stock price in a barrier option constitutes a dimension of time or not etc.

It'd probably have taken you less time to have humoured me and answered with the dimensions of the 2 examples i asked.

would still appreciate it :)
It would have taken me no time at all to not bother replying.

Did you actually read those links? "In applied mathematics, the curse of dimensionality refers to the fact that some problems become intractable as the number of the variables increases"

I don't mean to be short with you, but I mean, I googled your own original question "dimensionality of a pricing problem" and get plenty hits.

See: http://citeseerx.ist.psu.edu/viewdoc/download?doi=

"Many problems in mathematical finance can be formulated as highdimensional
integrals, where the large number of dimensions arises from small time
steps in time discretization and/or a large number of state variables."

Or alternatively: http://www.smp.uq.edu.au/content/lifting-curse-dimensionality

“High dimensional problems, that is, problems with a very large number of variables, are coming to play an ever more important role in applications. These include, for example, option pricing problems in mathematical finance, maximum likelihood problems in statistics, and porous flow problems in computational physics.”

Go back and read the wiki article on the curse again. Particularly the comparison of sampling a line, versus a hypercube. It is my impression that it may be helpful think of your problem in terms of geometric dimensions.. a dimension is an axis, (a variable, along which some value may be observed, ala an axis, NOT a data point representing an actual observation, ala your 30 days daily obs).

Side note: I do not believe that the variables must be stochastic in nature. And unless I'm wildly mistaken, your single geo original example would be a 1 dimension (assuming your brownian motion is a standard 1D brownian motion and is the only variable driving the prices)

Pleasant reading.

(Slightly apology. Bad day at work.)
ALso, try searching the wilmott forums. They tend to be a technical nature and may be of use. But they won't suffer poorly thought out questions, so make sure you do your homework before you flood them.