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Verde
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### Actuarial Pricing vs Financial Mathematics Pricing

QuoteOriginally posted by: Edgey@VerdeI don't disagree with your post, but you haven't argued against my point (that the actuarial approach doesn't need independence). Insurance companies do use independence to reduce their costs, but independence is not a required assumption for pricing. Actuaries do add costs on for reserves, but this is not part of the the expected cost under the actuarial approach. To be clear, my definition of the "actuarial approach" is to use historical probabilities to derive prices, rather than market implied probabilities. You may have a more broad definition.OK, I think I get it. I am viewing the issue from the point of view of an insurer who needs to meet obligations out of premiums collected. You see it from the point of view of an investor using the actuarial approach to derive prices, which is the correct approach in the context of this discussion.

numbersix
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### Actuarial Pricing vs Financial Mathematics Pricing

Last edited by numbersix on June 22nd, 2011, 10:00 pm, edited 1 time in total.

Alan
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### Actuarial Pricing vs Financial Mathematics Pricing

QuoteOriginally posted by: numbersixAdmittedly, the major conversion I am proposing, in which price absolutely replaces probability, leaves us in the later impossibility of modeling the dynamics of price. For how could we model its dynamics except through probability? But do we really need to model it, now that it is given by the market? Isn't the rule precisely constantly to recalibrate our models of the underlying dynamics to the market prices of derivatives? Better: are our derivative pricing models really models of the underlying dynamics or just risk-neutral pricing operators that allow us to capture a semblance of consistency between the instant prices of derivatives, with no idea of what will happen next apart from recalibration to the market update? Why indeed doesn't the market become a pricing theory of its own, THE pricing theory? Why do we need a theory for the market? Is it because the market is complex and we need to model it? Well, I say the market is simple, not complex. Just forget the crowd that constitutes it. Simply, the market is what gives the price (and the price process) of contingent claims. And if you're not happy calling the market a theory, then just call it a technology.I enjoyed this essay, and have absolutely no quibble with the thesis that, fundamentally, objective probabilities do not existin most market setups (ignoring lotteries/casinos, etc). What I don't see is the following. Go back to the 50's and 60's, whenlisted and well-developed derivative markets did not exist (except OTC), but well-developedstock and commodity markets did exist. It is self-evident that these markets were worthy of academic (meaning serious) study.So, indeed, people like Osborne and others collected price series and studied them. They found that, to a first approximation anyway,no matter what security you picked, the price change series behaved much like random walks. (Here by 'random walk',I mean it in the weak sense of simply uncorrelated price changes). The statistical properties of market priceseries beg for an explanation -- can you explain or even study these properties without invoking probability? (honest question)
Last edited by Alan on June 22nd, 2011, 10:00 pm, edited 1 time in total.

numbersix
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### Actuarial Pricing vs Financial Mathematics Pricing

Thank you. A quick reply is that I do believe in statistical laws and statistical regularity -- an indisputable law of nature, as I say in my 'essay'. However, the unwarranted move is to go from the statistical distribution to the random generator, from ex-post to ex-ante. True, this sounds like splitting hair, but the whole metaphysical divide between the empiricists who only believe in statistics and the realists who believe in the subsistence of generators lies in just this fine point.On the other hand, I have no problem considering that the market of contingent claims has always existed, even in the 50's and 60's, but that it is was virtual (or potential) then, not actual. (This doesn't make it less real.) As a matter of fact, I have no problem considering that the millennium-old problem of probability was waiting for the technology of contingents claims in its ultimate sophistication to find its answer.

Alan
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### Actuarial Pricing vs Financial Mathematics Pricing

OK, first sticking to ex-post, would you object to saying: well, we know that a data generator does not truly exist,but the data is reasonably consistent with certain generators? If so, then a data generator becomes merely anice short-hand for describing, to a first approximation, many of statistical properties of the series -- again, ex-post.Now, as to ex-ante, we probably agree that, regardless of what was said about our historical series, it'spotentially a qualitatively different ball-game when we look to future. Our statements have to become much moretentative. Nevertheless, if you were asked for your opinion that a 5 min. price change series of IBM,extending from today to a year from now, would exhibit similar low auto-correlation as it did historically, Iwill guess your answer would be yes. If so, what are your thought processes that support this opinion? (I ask because I am struggling to see how to discuss the random character of stock prices, toborrow Cootner's book title, in your framework.)
Last edited by Alan on June 22nd, 2011, 10:00 pm, edited 1 time in total.

numbersix
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### Actuarial Pricing vs Financial Mathematics Pricing

Agreed on the data being consistent with the theoretical posit of a generator, therefore the generator is a model, or a way of speech, or a shorthand for describing the phenomenon. As for the statistical regularity to persist in the future, I have no problem with that either, as this is the definition of regularity. But this is not ex-ante, this is believing in the stability of laws of nature. This is my thought process.Ex-ante would be to focus on the next individual 5 min. price change, I mean the one beginning right now, not on the 5 min. price change series, and to try to give a meaning to its probabillity distribution other than the ex-post fact that, as time goes by, it will belong to a whole population in which the statistical auto-correlation is indeed observed. To me, ex-ante is undissociable from 'single-case'.In short, we all write: dS = mu*S*dt + sigma*S*dZ (this is your 5 min. price change);What does that mean exactly?
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Alan
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### Actuarial Pricing vs Financial Mathematics Pricing

OK, well I am surprised to hear you say you believe in the "stability of the laws of nature", as I thought we would agree thatthere is fundamentally no 'stable law' that describes the upcoming IBM price series over the next year. But, if I take youat your word, then there seems to me a potential contradiction here. If you can elaborate on what the putative law is, then I suspect it will lead to some probability-type statements about the single case event of the next IBM 5 min price change. p.s. To answer your question, if it's not rhetorical, I would say dS = sigma(t) S dZ for the 5 min price changeis simply a very useful model for the typical investor (including myself) without some insider knowledge of the underlying firm.It obviously has a precise mathematical meaning and leads to probability statements about S(t+5 min)-S(t).It also leads to useful normative prescriptions about how to approach investments, how not to waste moneyneedlessly on short-term trading, and much else. Certainly, it's not a law of nature in the sense of physicallaw and is easily refuted if taken seriously on that basis. It also embeds the notion that, for many stocks, wehave excellent price continuity during the NYSE session*. So, as you say, it's a 'way of speech' -- a very helpful mental model for thinking about the unknowable future and forming judgments about it. *[Of course, continuity for a lattice process with penny increments strictly meansthat each tick of DeltaS will be $0.00 or$0.01. This is not quite true for IBM, asthe trade ticks are not that small, but if IBM split to say \$20, it probably would be].
Last edited by Alan on June 22nd, 2011, 10:00 pm, edited 1 time in total.

Marsden
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### Actuarial Pricing vs Financial Mathematics Pricing

numbersix, while admitting that I had to refresh my coffee a couple times while reading your essay, and therefore that I may have dropped some of your lines of reasoning inadvertently, I think you may be missing the forest for the trees.(Let me say at the outset that I think probability is always subjective; I think it's likely that this will come up at some point.)You seem, from my perspective, to accord prices far more "reality" than is warranted. In many respects, prices are no more real than are shadows: they have no mass; they are not subject directly to laws of physics such as we have developed them; etc. I think prices are very well considered in the way that Wittgenstein considered "meaning;" meaning is defined by use.What is the use of a price? Prices are not stand-alone quantities that support nothing more than a new-fangled regime of gambling; they represent economic realities. In particular, they tell us something about the economic circumstances that brought them to wherever they are, and they direct our own economic activities by giving us insight on the rewards for different undertakings.But how can this relationship with economic reality function without "objective probabilities?" (And here is where my belief that all probability is subjective comes up: "objective" as used in "objective probablility" is, as I understand it, one's best guess at the probability distribution of future outcomes, independent of other considerations that may enter into prices such as utility, etc. "Objective probabilities" are subjective because they are not "God's own true probability distributions," but rather they are conceived separately by individual agents based upon the information -- and possibly the misinformation -- that each agent possesses. I hope that doesn't disagree with anyone else's understanding, but I thought in any case that I'd be very clear about it.) Maybe it can, and certainly "objective probabilities" are not the only inputs affecting prices, but I can't clearly see how prices would retain their economic meaning and function without any consideration of "objective probabilities."
Last edited by Marsden on June 22nd, 2011, 10:00 pm, edited 1 time in total.

numbersix
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### Actuarial Pricing vs Financial Mathematics Pricing

Alan,Just to be clear -- and I am sorry for any confusion -- I just followed on your IBM example because I thought you were only interested in exhibiting the passage between ex-post and ex-ante, between the statistics and the data generator that is merely a shorthand way of speaking of the statistics. The law of nature that I referred to in that instance was just the statistical regularity.However, if what you really had in mind was the IBM stock as such, qua denizen of the market, then of course I don't believe in any law there, because I believe the statistics of prices are not stationary. On the contrary, I am firm believer that the market is a nest of Black Swans. (And I don't know what happened in the 50's - 60's for the distribution to seem stationary then.)Still, even if there were stationarity and the putative statistical law was invoked (say we are not talking about markets but about mortality), certainly, as you say, this will lead to probability-like statements concerning the next single case event as this is just a statement and we have agreed that probability is just this way of speech, but the whole point is precisely to see what this statement could possibly mean other than, blah blah, the next tick will be part of statistical population with the corresponding distribution, etc. In the end, this, once again, refers to the population, not the next tick as such.I quickly dismissed it in my 'essay', but the real pressing case for single-case objective probability (or propensity) is quantum mechanics. Here, truly, there is something going on concerning the single-case as such, that is both random and not necessarily related to a population. Quantum mechanics is really what pressed the case of irreducible single-case objective probability. However, as it happens, the notion of generator and possible state is inadequate here, because quantum indeterminacy is not about probability, it is about something deeper.
Last edited by numbersix on June 22nd, 2011, 10:00 pm, edited 1 time in total.

Fermion
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### Actuarial Pricing vs Financial Mathematics Pricing

QuoteOriginally posted by: MarsdenYou seem, from my perspective, to accord prices far more "reality" than is warranted. In many respects, prices are no more real than are shadows: they have no mass; they are not subject directly to laws of physics such as we have developed them; etc. I think prices are very well considered in the way that Wittgenstein considered "meaning;" meaning is defined by use.What is the use of a price? Prices are not stand-alone quantities that support nothing more than a new-fangled regime of gambling; they represent economic realities. In particular, they tell us something about the economic circumstances that brought them to wherever they are, and they direct our own economic activities by giving us insight on the rewards for different undertakings.But how can this relationship with economic reality function without "objective probabilities?" (And here is where my belief that all probability is subjective comes up: "objective" as used in "objective probablility" is, as I understand it, one's best guess at the probability distribution of future outcomes, independent of other considerations that may enter into prices such as utility, etc. "Objective probabilities" are subjective because they are not "God's own true probability distributions," but rather they are conceived separately by individual agents based upon the information -- and possibly the misinformation -- that each agent possesses. I hope that doesn't disagree with anyone else's understanding, but I thought in any case that I'd be very clear about it.) Maybe it can, and certainly "objective probabilities" are not the only inputs affecting prices, but I can't clearly see how prices would retain their economic meaning and function without any consideration of "objective probabilities."Yes! What are prices? I claim they are aggregate market estimates of present value perceived by market participants. They are not the present value itself, but they imply that market participants themselves have a probability distribution in mind not just in the future but also at the present instance. One does not have to believe in objective probability or even that the distribution is knowable to recognize that an implicit price distribution exists in the market both now and in the future as a result of the aggregate behavior of market participants and that distribution collapses to a unique value at the moment of a trade.QuoteOriginally posted by: numbersixHowever, if what you really had in mind was the IBM stock as such, qua denizen of the market, then of course I don't believe in any law there, because I believe the statistics of prices are not stationary. On the contrary, I am firm believer that the market is a nest of Black Swans. (And I don't know what happened in the 50's - 60's for the distribution to seem stationary then.)At the current instance, with all available information known (by definition) there are no black swans to affect the current distribution of possible prices out of which the market selects a price. As regards the future, one can conceive of a future of black swans so extreme as to make the mean of the future distribution completely unpredictable, but leaving its shape unaffected -- or at least a fairly stable function of time. When it comes to valuing derivatives, risk neutrality tells us that we don't care about the expected value -- and by implication nor do we care about those black swans -- only the shape of the distribution matters and that is a product of trader behavior combined with the aggregate magnitude (but not direction) of black swan events.
Last edited by Fermion on June 23rd, 2011, 10:00 pm, edited 1 time in total.

numbersix
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### Actuarial Pricing vs Financial Mathematics Pricing

QuoteOriginally posted by: Marsden1) You seem, from my perspective, to accord prices far more "reality" than is warranted.2) Prices are not stand-alone quantities that support nothing more than a new-fangled regime of gambling; they represent economic realities.3) "objective" as used in "objective probability" is, as I understand it, one's best guess at the probability distribution of future outcomes.4) I can't clearly see how prices would retain their economic meaning and function without any consideration of "objective probabilities."In other words:- The only thing that is real is the present economic reality. - When this reality is future, it is not real; it is just possible, therefore it is dealt with by probability.- Probability is not real; it doesn't replace the missing reality by some real random generator; it takes place in the minds of people.- Only in the last instance do prices enter into play. - Prices are even less real than probabilities, because they consist only of meaning and use, and the only meaning they derive is through probabilities.Here is an alternative set of propositions:- Contingent claims are not possibilities; they are real sheets of paper (also called contracts) that translate into real money, only contingent, in the future reality.- What they translate into, today, are prices -- real money too, no less contingent mind you. - It is the same contingent claims that we have today (and exchange for prices) and will have tomorrow (and exchange for payoffs). Possibilities, by contrast, can dramatically change or be revised (after Black Swan events).- Money is accountable; probability isn't.- I wonder whether the future economic realities, as seen from today, can be translated otherwise than by future contingent payoffs.- I wonder whether the present economic realities, as felt today, can be translated otherwise than by the present prices.Note that derivative pricing theory is based on probabilistic models of prices, not of abstract states of the world or of the economy. That's why it evolved into a real technology (involving software and listed exchanges) while economic theory didn't and hardly left the stage of game theory.All that remains to see is that derivative pricing theory, now understood as a technology and no longer as the candid textbook theory everyone thinks it is, closes itself off on prices, via calibration and recalibration. Its textbook use of probability, as I have said, is only lip service to the actuaries.
Last edited by numbersix on June 26th, 2011, 10:00 pm, edited 1 time in total.

Fermion
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### Actuarial Pricing vs Financial Mathematics Pricing

numbersix:1. Do you agree that the future consists of possibilities?2. Do you agree that some possibilities are more likely than others? E.g. if a stock has a price 100 today then it is more likely that the price will be 100 tomorrow than that it will be 1000 or 0?If you answer yes to both these questions, 3. why does it matter that events can change possibilities? Doesn't that just mean that the more or less likely status (i.e. relative probability) changes conditional on new events?If your response to (3) is that we can't forsee the black swan event and so cannot assess the probability, thena. In what sense was it possible to say that one price was more likely than another tomorrow in the first place?b. Why can't we simply say that there is a probability of a black swan event and add it in to our estimates. And if there are black swans on top of black swans recursively, why can't we add those in too?In other words, why cannot we consider black swans to be merely extremely unlikely events and (say) put a maximum probability on their occurrence rather than throwing up our hands and saying "black swans destroy probability" -- which seems to be your line.Of course there are some black swan events which genuinely could destroy not just future probability but markets themselves (we even anticipate knowable extreme events -- that aren't even black swans -- that can do that such as nuclear annihilation, asteroid collision, Yellowstone eruption and so on) but, in that case, don't we just say "well if that happens we're all dead anyway, so we might as well continue as if they won't happen" (i.e. treat their probability as miniscule and die unavoidably if we're wrong). An alternative way of stating this is that we can necessarily concern ourselves only with events we can anticipate since, by definition, we cannot do anything about events we cannot anticipate.

Marsden
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### Actuarial Pricing vs Financial Mathematics Pricing

QuoteOriginally posted by: numbersixIn other words:- The only thing that is real is the present economic reality. - When this reality is future, it is not real; it is just possible, therefore it is dealt with by probability.- Probability is not real; it doesn't replace the missing reality by some real random generator; it takes place in the minds of people.- Only in the last instance do prices enter into play. - Prices are even less real than probabilities, because they consist only of meaning and use, and the only meaning they derive is through probabilities.The only thing I immediately don't agree with is "the only meaning they (prices) derive is through probabilities." There are also things like utility that potentially affect prices but that are not matters purely of probability.QuoteHere is an alternative set of propositions:- Contingent claims are not possibilities; they are real sheets of paper (also called contracts) that translate into real money, only contingent, in the future reality.- What they translate into, today, are prices -- real money too, no less contingent mind you. - It is the same contingent claims that we have today (and exchange for prices) and will have tomorrow (and exchange for payoffs). Possibilities, by contrast, can dramatically change or be revised (after Black Swan events).- Money is accountable; probability isn't.- I wonder whether the future economic realities, as seen from today, can be translated otherwise than by future contingent payoffs.- I wonder whether the present economic realities, as felt today, can be translated otherwise than by the present prices.I don't see how from this set of propositions economic reality that exists prior to prices based upon it can be recognized; if tomorrow a huge new oil deposit is discovered, how is its value, as reflected in the prices of whatever ownership exists for it, determined?I guess that basically I see your system of belief as lacking a creation myth ...QuoteNote that derivative pricing theory is based on probabilistic models of prices, not of abstract states of the world or of the economy. That's why it evolved into a real technology (involving software and listed exchanges) while economic theory didn't and hardly left the stage of game theory.Here you seem to envision creating, essentially, a system of physical laws that apply to prices. I don't know if that's wise. Earlier, I noted that prices are in some ways no more real than are shadows. If we were to try to create a system of physical laws that applies to shadows solely through the method of observing shadows, I think we'd end up with a mess. It would only be by the insight that shadows are created by a light source, a light blocker, and a reflecting surface, and that each of these will be composed of matter and subject to physical laws regarding matter that we might have a reasonable hope of creating a derivative set of "physical laws" that apply to shadows.Do you think I'm raising a false parallel, and if you do, why?QuoteAll that remains to see is that derivative pricing theory, now understood as a technology and no longer as the candid textbook theory everyone thinks it is, closes itself off on prices, via calibration and recalibration. Its textbook use of probability, as I have said, is only lip service to the actuaries.I think that if a price were readily available for every conceivable thing, your system would hold together perfectly. But I think there will always be something with economic significance for which no price is available; possibly that is even a proposition that can be proven. Without having given the matter too much thought yet, I think in these situations your system will want a creation myth like I noted above. What happens with your system when it runs into needing initial and boundary conditions?