# Talk:Mathematical finance

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## Notes

Mathematical finance is the foundation of financial engineering or computational finance. It appears though that financial engineering FE is more concerned about derivative pricing and hedging using applied math and computational methods. After examining the various masters degree level programs in mathematical finance and financial engineering/computational finance available in the US, it appears that mathematical finance is defined by a heavier emphasis on mathematics vs. finance courses although finance problems are central to the various theories. Some programs are extremely heavy on probability and stochastic calculus. I surmise that if one pursues a PHd in the field, then FE and MF could be seen as one and the same.

Rather than this useless back and forth editing, could we instead discuss the notability and reasons to include the contested external links? finmath.com looks like just an amazon affiliate with no notable references to its pages...am I missing something? What external links should this article include instead of or in addition to the links already listed? Flowanda | Talk 09:47, 10 January 2008 (UTC)

Not sure what's to discuss. It's WP:SPAM. I'm calling in the anti-spam cavalry. Ronnotel (talk) 12:05, 10 January 2008 (UTC)
finmath.com has now been blacklisted as spam. Ronnotel (talk) 16:24, 10 January 2008 (UTC)

## Q versus P, semantic bracketing

Hi all. In the History: Q versus P section, the first sentence says

There exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing and risk and portfolio management.

Does this mean [derivatives pricing and risk] and [portfolio management], or [derivatives pricing] and [risk and portfolio management]? I assume it means the latter. A judiciously place comma would certainly make it clearer, ie

derivatives pricing, and risk and portfolio management.

Have I got that the right way round? May I make that change?

Thanks. Tez (talk) 18:55, 24 September 2014 (UTC)

Ah, I see the distinction in the subsection headers. I've been bold and made the change. Thanks, Tez (talk) 19:02, 24 September 2014 (UTC)

## nomination for deletion

hello,

i am having some serious difficulties following the instructions for deletions, as i have had no luck in creating the deletion discussion page https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion#Nominating_article.28s.29_for_deletion

there can be many reasons i could give for deletion:

1. the Ito Integral, a tool central to this field, is not a real integral. this claim is justified by the fact that the measure used in this integral, a so-called Martingale measure, does not satisfy the criteria of a sigma-algebra 2. the (varying) assumptions of continuity that the data cannot meet. that is, no data collected in this field is truly continuous. they are not measurements taken from nature. rather, they are values generated by some underlying (unknown) regime, similar to how we view hidden variables in a hidden markov model, except, again, the data being used to enforce these assumptions, is not collected from nature.

these are two easy ones i am sure someone can argue, so let's get 'er started!! GL HF ALL 174.3.155.181 (talk) 04:30, 3 April 2016 (UTC)

(1) If you don't want to call it an integral, fine. It's well defined as a linear functional on a space of stochastic processes or as a Moore-Smith limit of Riemann sums over the directed set of partitions of an interval.
surely you are aware that there are more conventional definitions of limits used in mathematics, right? often, one shouldn't need more than the (ε, δ)-definition of limit. in any case, anything google pulls up with this "moore smith limit" involves topology. topology requires some geometric structure that the input data cannot provide. also, you would notice that this same limit introduces concepts of things like open sets. it is curious that risk-neutral measure does not demonstrate the same properties
(2) Please note the distinction between model and reality.
it doesn't matter when you say "model and reality" because that's just a good sound bite (:P). what puzzles me is the need for any measure other than the Lebesgue integral when we assume there are movements over time. Henri Lebesgue's integral should cover any value on the [0,1] interval, thus allowing the integral to capture "rapid" movements that humans could not. assuming we had some data collected over a fixed time interval, i do not see how the ideas from risk-neutral measure supercede those of the lebesgue integral because the latter should be able to accommodate all possible values whilst providing clear geometry that everyone can see.
here's another curious thing. i noticed that after an expert in the area i'm discussing passed away (https://web.archive.org/web/20100324055343/http://www.nationalpost.com/news/canada/story.html?id=2439178&p=2), that user:VanishedUserABC created a page called Probability measure to distinguish it from the Lebesgue measure. not only is this suspicious, but i feel compelled to call in a few administrators to investigate this. there is no need to distinguish a probability measure from the lebesgue measure. i find it curious that the aforementioned expert, who designed a method to demonstrate the existence of the lebesgue measure, was absent at the time this page was created. it's also curious user:VanishedUserABC padded their talk page with erratic edits, as to "cover tracks" to the best of their ability.
here's the proof that roweis&i's contribution successfully uses the lebesgue measure to recover measurements over time, all by treating fMRI data as time series http://imgur.com/bhAQPCK
i would be happy to provide you a full fMRI file that you can visualise, using the program available at http://medinria.fr
i believe that this image shows the geometry of numbers, a field founded by the great hermann minkowski. so if our contribution successfully uses the lebesgue measure to endow fMRI data with structure, similar to what is being claimed by mathematical finance, surely both cannot be true? that is, unless, there is some clear way to impose some geometric assumption on financial markets data.
listen, much of the premise of this field relies on misinformation. both risk-neutral measure AND Martingale (probability theory) link to probability measure because they are not measures. look at the page Measure (mathematics). it generalises the concept of "length, area, and volume" in higher dimensions. notice how those have geometry?
note: stochastic processes is NOT Mathematical finance. you must learn to distinguish between the two, because the latter uses the former, but not vice versa.
you tried to argue with me that the field is legitimate, and cannot even prove the measures used have legitimate geometric properties. are you an expert, or someone who was ensnared into a 6 figure student education debt, with high hopes of the "world of high finance"? i am an expert, i have demonstrated my full understanding of measures, and now i expect the same from you.
*show* me that the measures referred in mathematical finance are legitimate by providing some sort of geometric description. this is a *requirement* for measures because length, area and volume of an object has a clear, measurable, geometric properties. it may be complex geometry in 3d, but it is still measurable. i'm only asking for a case in 2d, which should be easy (assume the x-axis is time interval. use any amount of time points that you want. treat each object/security with a price as a separate line on this axis). 174.3.155.181 (talk) 06:51, 3 April 2016 (UTC)
This page should be kept. I'm going to just respond briefly at this time because I don't have the time for a more detailed response:
stop making cowardly claims like "i don't have the time". what are you so busy doing, mr PhD in financial engineering? i have heard this excuse from "experts" in your field many times, and it's often a cowardly way to mount a half-hearted defense followed by a retreat. be a man and and accept the flaws and garbage that you spent 10 years learning. now look at you, you're indoctrinating new kids into a field that has no basis because that's what you must do to survive. have you any shame? you should quit wikipedia like User:VanishedUserABC did once he learned of my contribution, coward.
1. A probability measure ${\displaystyle \mu }$ on a space ${\displaystyle X}$ is a measure with ${\displaystyle \mu (X)=1}$. It is helpful to interpret the result of the measure to be length or volume as the Lebesgue measure in Euclidean space, but that is not the only measure that exists. But yes, because of non-negativity and countable additivity you will recover an idea like volume in whatever space you are working.
none of what you said makes sense. "because of non-negativitity and countable additivity you will recover an idea like volume"? what do you mean "like volume"? if it was volume, then it would be the lebesgue measure!! let me play along with your word salad, just to show you how terribly weak your arguments are: SHOW ME THE non-negativity and countable additivity properties being satisfied on the interval upon which these values apparently lie!! that is, given a set of numbers that represent the value of the underlying bond/security/etc, show me that the interval over time these values exist over satisfy the properties of measure.
By "like volume", I mean to compare the idea of a measure to volume. Just as you asked for previously. As for your question: for which measure are you asking about? For a general probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$? The interval may not be a well defined statement on ${\displaystyle \Omega }$. If you wish to stick with the Lebesgue measure, you fall into a trap when you want to work on an infinite-dimensional Banach space (infinite-dimensional Lebesgue measure). As for the application to finance: the value of an asset is a random variable, i.e., a measurable mapping ${\displaystyle X:\Omega \to \mathbb {R} }$. The notation ${\displaystyle P(X\leq x)}$ actually is interpreted as ${\displaystyle P(\{\omega \in \Omega \;|\;X(\omega )\leq x\})}$ for instance. Thus if ${\displaystyle \Omega }$ is not a lattice space the term "interval" has no meaning. Zfeinst (talk) 20:27, 3 April 2016 (UTC)
how many times have i told you to stop deferring to unproven definitions created by questionable users? working with the lebesgue measure does not introduce any trap that you speak of. as i mentioned in my response on Talk:probability measure, the computer is inherently discrete. thus the state space is always finite. the set of values that can be taken on in a banach space on a 64 bit computer is 2^64 -1. i assure you that this is not "close enough" to infinity because 2^64 - 1 potential values just improves the precision. infinite dimensional banach spaces (when referring to computation) would only be true in practice with quantum computers (qubits should take on any value on the [0,1] interval which would be far far far far greater than 2^64 - 1. infinite uncountability)
how many times must i tell you that your area isn't mathematically rigorous? if you were such a hot shot, use the definition of measurable function, as i've asked. i shouldn't have to keep asking you to do this. use any asset you want, plot its values over a reasonable number of time points (100 evenly-spaced-over-time measurements would do. i.e. price on day 1, day 2,... day 100, or price at minute 1, minute 2, ..., minute 100) and demonstrate to me that the assumption of continuity between each of the prices is legitimate. essentially you cannot, because the laws of nature do not apply to asset pricing. i will come back to this claim when i address your disrespect of the great Isaac Newton, the greatest mathematician who ever lived, and his laws.
2. Further, making the claim that the martingale measure is not a sigma algebra is meaningless, you would have a probability space of which the probability measure is the martingale measure, the measure and the sigma algebra are two (interrelated but distinct) components of the space.
are you listening? the martingale measure is not a real measure. the martingale measure is predicated on probability measure, which i stated was created after the death of *THE* definitive expert in this field had committed suicide. listen to what you're saying "you would have a probability space of which the probability measure is the martingale measure". NO, when you define something as a measure, they ALWAYS satisfy the same properties. i've asked for a demonstration using a very simple set of numbers produced over time (that represent the object's "value").
Yes any measure satisfies 3 properties (and a probability measure one additional property). These are recounted in the page measure (mathematics). The probability measure ${\displaystyle Q}$ satisfies exactly these properties, and is such that the discounted price of the security is a martingale. Do you wish for a proof of the fundamental theorem of asset pricing? Zfeinst (talk) 20:27, 3 April 2016 (UTC)
tell me, did you feel smart when you thought i'd want a proof of a self-referencing theorem like fundamental theorem of asset-pricing. it is so laughable that you actually even tried to present this to me, as if it's some equivalent to the fundamental theorem of calculus. do you actually believe language like "arbitrage-free" (or even arbitrage, for that matter) have legitimate mathematical basis? it does not. finance is not mathematics. what do you not understand? i really must say you are in need of psychological evaluation if you feel your beloved fundamental theorem of asset pricing deserves to be on the same ground as the fundamental theorem of calculus. the fact that the former is named similarly to the latter suggests a high amount of sophistry.
just think, for one second, about what you tried to assert. you do realise that the fundamental theorem of calculus shows that integration and differentiation (two techniques ABUSED by your so-called field) are linked, right? you guys haven't even shown basic properties of infinite uncountability on the interval (over time) that your field claims to analyse. the numbers analysed by your discipline are inherently discrete/rational. they are measurements that are generated arbitrarily, and output by either humans or computers (the latter can introduce quite a few additional decimal places, thereby increasing the perceived "sophistication" of your discipline
3. To say the Ito integral is not a real integral does not change the fact that stochastic calculus is mathematically rigorous and that this is the accepted terminology.
first of all, if the Ito integral is not a real integral (which you failed to dispute, curious because the previous poster made a lame attempt). your argument can be interpreted as "even if the ito integral is not real (a central tool used in finance), stochastic calculus is real". i am not disputing that elements of stochastic calculus are valid. however, the validity of the theorems and methods that comprise stochastic processes were borne out of curiosity about nature's behaviour over time, and how we could measure it. i am well versed in this area (http://arxiv.org/abs/1407.8392 http://web.archive.org/web/20150923213902/http://webdocs.cs.ualberta.ca/~gagans/) so i urge you to tread very, very carefully.
We are not disputing your statement about a "real integral" because you don't define what that means. Is the Riemann integral "real"? The Lebesgue integral? What mathematical properties must an integral satisfy? We granted you that statement because it is irrelevant to the point at hand. If the Ito integral were termed the "Ito function" instead it would have no difference in the mathematical results, so granting you that point is of no consequence. Thank you for the warning, I am treading carefully... I'm not sure what your dispute is though? Stochastic calculus is valid, and is being applied to finance, and then...? Zfeinst (talk) 20:27, 3 April 2016 (UTC)
of course the riemann integral is real. you take sums via the trapezoid rule, which then provide the area under the curve. i never said, nor did i imply, the integrals used in vanilla calculus are not integrals (Riemann-Stieltjes integral Riemann integral Lebesgue-Stieltjes integral)). to suggest that the integrals your field deploys are on the same level is wrong because
the riemann integral inherently uses the concept of the rectangle rule and/or the trapezoid rule. by deploying either procedure, these rectangles/trapezoids become infinitely-small (measure 0 on the x axis) with heights equal to f(t) (where 't' is the value, aka height, taken on by f(x) at time t). notice how infinite uncountability, again, implicitly plays a very large role in these integrals? the value output by is riemann integral implicitly in the limit, because perfectly determining the area under the curve would require an infinite number of rectangles (of different heights) whose areas then sum to the value given as output.
contrast these conventional integrals to the ito integral, which does not systematically break down AND analyse the interval in the same way for EACH "broken up" interval. for example, look at Ito calculus#integration by parts. notice how the measure is different after "breaking up" the integral (dY and dX)? for riemann or lebesgue integration, no such swapping occurs.
the excuse of quadratic variation, which wishes-away the clear shortcomings of this purported "integral", further demonstrate that your so-called field is fraudulent. the entire page is dedicated to financial analysis. tell me, is functional Magnetic Resonance Imaging data so inferior to financial data that we have no need for methods to account for error the approximated (lebesgue) integral? we do, but we abide by rigorous mathematics such as Regularization, Bias-variance tradeoff, or loss functions (which are quadratic) to guide our analyses. in contrast, your field hijacked a common mathematical term (quadratic variance) and monopolised it for your shyster area!! have you guys any shame?!?!
4. It is well understood that the models used do not perfectly capture the underlying reality, but neither do, e.g., Newton's laws, yet those are still useful models for most circumstances. If you prefer, there exist jump diffusion or other more sophisticated models rather than just the geometric brownian motion discussed in this page.
are you seriously trying to compare the great sir isaac newton's work to your area? are you actually trying to insinuate that the work you're defending is on the same level? this is hilarious. first of all, all three of sir isaac's laws have *geometric* meaning that we can depict using free body diagrams. stop deflecting and show me what i was asking, which was demonstration of the properties of these purported probability measures (again, a page created shortly after the death of an expert in the area. the same user who created this page then FLED upon hearing of my contribution, which USES/DEMONSTRATES the lebesgue measure using fMRI data!). using the values of a bond or security over some well-defined time interval, complete with a picture etc should be sufficient to show "something like volume" or area (notice how i'm using your own words against you. we cannot interpret volume or area outside of euclidean space [assuming cartesian coordinates]).
First, Wikipedia is not a source for original research (Wikipedia:No original research), so without a citation/reference for your claims on the Lebesgue measure, it is not relevant to the discussion at hand. Second, what properties exactly are you curious about? The fact that the martingale measures satisfy the definition of a measure (mathematics)? Or what? A geometric interpretation is going to depend on the underlying measurable space. Do you want a construction of the martingale measure for a single-period binomial tree model? See Risk-neutral measure#Example 1 - Binomial model of stock prices. The think "like volume" would be the weight that we give to the up/down motion of the stock price. No it is not actually a volume, but if you insist on geometric interpretation of probability, then it is "like volume" in that it is the outcome of a measure. Zfeinst (talk) 20:27, 3 April 2016 (UTC)
you are continually referring to ideas from your own field. this is what you'd call self-citation. none of what you're claiming has been accepted by any mathematician. it may have been accepted by greedy people seeking to enrich themselves, who had some math knowledge, but it definitely is not up to snuff.
newton's laws do not need citations, just common sense. to homogenise comparisons, if i had fMRI data collected over 100 uniformly-spaced time points, each of those time points contain measurements over LxWxH voxels (3d). now, since these measurements are collected from *nature* (unlike your financial data, which, at the risk of sounding repetitive, is arbitrary and discrete) we know Newton's laws provide bounds on what can occur in time elapsed between two time points at the same location. that is, assume we measure location (0,1,0) at time t_1 and t_2. sir isaac newton's laws provide rules that allow us to infer the motion between these two points. for example, we know that macroscopic motion (such as the neuron's action potential) itself cannot exceed the speed of light. reading up on molecular mechanics may help facilitate your understanding of what happens.

5. I haven't actually seen an argument for the deletion of this specific page. The arguments are for the deletion of probability measure and instead redirect to measure (mathematics) but this is not the location for that debate, that should occur on those pages themselves. The second argument is that the historical modeling is oversimplified, but this is an active field of research, so to say that the first widely used model does not fully capture reality is tantamount to saying any field that did not have a model that did not match all data from day 1 is unworthy of existing. Zfeinst (talk) 12:56, 3 April 2016 (UTC)
there is a difference between "quantitative" and "mathematical" when prepending the terms in front of "finance". in the former case, you are studying the field of "finance" from a quantitative perspective. under such a view, i could tolerate half the nonsense you're espousing because it's not mathematics (there is no axiomatic approach to your field. there is no consistent, methodological deference to the axioms of measure etc in the same way as people in the field of mathematical analysis (my area of expertise). thus, one must question whether it is MATHEMATICS (a rigorous study) or just the study of QUANTIFICATION (numbers are but ONE part of mathematics).
I agree there is can be a difference between mathematical finance and quantitative finance. What axioms are you looking for? Are you therefore arguing that this page would be better suited to be split with one page for math finance and another for quantitative finance (which currently redirects to math finance)? Or is your proposal to rename the page quantitative finance? Your statement that something is "one part of mathematics" is not an argument. Stochastic calculus is "one part of mathematics", but it is still mathematics by your own admission. Zfeinst (talk) 20:27, 3 April 2016 (UTC)
move the entire field to quantitative finance and i will be happy with that. i do not like the usage of the term mathematics. it's that simple. you guys are free to procure any analysis you want, but you can't call it mathematics. is it quantitative? yes. does it use elements of mathematics? it tries to, but these "noble" attempts do not make it mathematics.
i appreciate your flailing attempt at defending your field, but now i've shown my skills and my expertise. i have shown the ability to measure natural phenomena using rigorous mathematics, evidenced by this work here http://arxiv.org/abs/1407.8392 (i am very grateful to the reviewers/co author/editor [the latter of which works for RAND] for assisting with this journey). please do not put forward another half-hearted attempt again. you would be better served to just let the next person try to defend the field.
Looks like a very interesting paper, I'll be sure to give it a thorough read at some point. Though based on the abstract I'm unsure how machine learning and baseball are relevant to the topic at hand. Zfeinst (talk) 20:27, 3 April 2016 (UTC)