Page 2 of 4

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 15th, 2011, 6:01 pm
There is very little written on this topic that wasn't either written by an exclusive actuary (who doesn't fully appreciate the finance) or written by an exclusive quant (who doesn't fully appreciate the actuarial science).You might look at "Variable Annuities" published by Risk Books for a good survey of the guarantees, issues and practical techniques.

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 16th, 2011, 2:25 pm
Actuaries would value a stock as its discounted mathematical expectation under the 'objective probability measure', whatever the 'objective probability measure' means. I, personally, think it is meaningless.This actuarial value would come out greater than the actual market price. Put differently, financial investors in the stock want to pay less than the premium that would make them 'break-even on average and in the long run' (whatever that means -- again, I think this crucial actuarial concept is devoid of meaning in the market), i.e they wish to earn a risk premium, because of the riskiness of the stock.If you insist on expressing the financial price of the stock as an actuarial value, i.e. as its discounted mathematical expectation, you then have no choice but to change the probability measure. You end up with the so-called 'risk-neutral' probability measure, where the financial price of the stock is now equal to its discounted expectation. The objective probability measure (whatever that means) and the risk-neutral measure have to be equivalent, i.e. the risk-neutral measure has to assign a zero price to the events that are impossible (or have zero measure) in the objective probability.Note that nothing forces you to relate the market price of anything to an underlying 'objective probability', or even to use the concept of probability in the first place. A price is just a price and a trader is not an actuary -- he doesn't break even in the long run. If there is no such thing as an objective probability measure ruling the stock, there is no 'changed probability measure' either. Also note that there is an infinity of such risk-neutral probability measures. All that is required is that the stock and the risk-less bond should be priced correctly.Non arbitrage theorem then yields that any contingent claim has to be valued under the same risk-neutral probability. In BSM, it suffices to have the market price of one option to nail that probability measure. In more complex stochastic volatility or jump-diffusion models, it takes calibration to a few more derivatives.In sum, for those like me who believe that objective probability is an ill-defined concept or even that it doesn't exist, the market is a blessing. For prices do exist; and no arbitrage imposes that they be formally expressed as discounted mathematical expectation. This is just a formal operator, also called 'pricing kernel'. It is not a 'changed' probability measure: changed from what?

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 18th, 2011, 2:51 am
I expect that you merely miss-wrote, but I believe the statementQuoteThe objective probability measure (whatever that means) and the risk-neutral measure have to be equivalentis false.Also, regardingQuoteNote that nothing forces you to relate the market price of anything to an underlying 'objective probability', or even to use the concept of probability in the first place.While this may be true when understanding the term "force" in the strongest possible sense, I think that for investors (as opposed to extremely pure traders), the desire to make a profit forces one to relate the market price of a security to what you think it will be in the near future, which I guess is roughly what you mean with "objective probability."Admittedly, I'm an actuary.

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 18th, 2011, 8:57 am
Hi Marsden,The first statement is obviously false, for how can something that exists (the pricing kernel) be equivalent to something that doesn't (the objective probability)? I did not mis-write; I was merely speaking from the point of view of the textbook presentation of risk-neutral probability which proceeds as follows: - First the objective probability, where it is assumed that investors would break-even on average if they valued assets as their discounted mathematical expectation (by the way, is there another meaning to 'objective probability' than 'statistical probability'?).- Then the change of probability measure in which investors now price the assets, i.e. exchange them in a market where anything goes and 'exchange value' is no longer related to any 'long run'.Being admittedly a pure trader, I only value contingent claims (a term I prefer to 'assets') relatively to other contingent claims by reverse-engineering the pricing kernel from the latter. So the only price movement that I expect and that triggers my 'investment' is that a certain asset, that I find mispriced relatively to the others, might undergo the movement that corrects the mispricing. Obviously, no probability governs this correction, as the correction is literally what falls outside my probabilistic model, not to mention that my probability is risk-neutral to begin with, hence implies nothing as far as movement in the real world is concerned.Not to mention that the 'mispriced' asset might never chance to correct its mispricing and that I might end up recalibrating my pricing kernel as a result, taking now this 'invalid' price as a new 'valid' input for my new model.

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 19th, 2011, 10:53 pm
"Doesn't exist" isn't really correct: aren't there potentially thousands of "objective probabilities" for every contingent claim?

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 20th, 2011, 12:01 am
QuoteOriginally posted by: MarsdenI expect that you merely miss-wrote, but I believe the statementQuoteThe objective probability measure (whatever that means) and the risk-neutral measure have to be equivalentis false.I will bet a nickel that Marsden is reading "equivalent" to mean "the same", while, of course numbersix means"equivalent" in the textbook/measure theory sense. If I'm right, then, for Marsden's benefit, the latter usage is that the two probability measures merely agree on what events are possible/impossible. So, equivalent measures merely agree that any stock price in [0,infty), say, is possible and any negative price is impossible, and that's it.

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 20th, 2011, 12:24 am
QuoteOriginally posted by: ClosetChartistThere is very little written on this topic that wasn't either written by an exclusive actuary (who doesn't fully appreciate the finance) or written by an exclusive quant (who doesn't fully appreciate the actuarial science).One paper that balances the two is Modern Valuation Techiques (2001)

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 20th, 2011, 11:00 am

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 20th, 2011, 11:36 am
QuoteOriginally posted by: AlanQuoteOriginally posted by: MarsdenI expect that you merely miss-wrote, but I believe the statementQuoteThe objective probability measure (whatever that means) and the risk-neutral measure have to be equivalentis false.I will bet a nickel that Marsden is reading "equivalent" to mean "the same", while, of course numbersix means"equivalent" in the textbook/measure theory sense. If I'm right, then, for Marsden's benefit, the latter usage is that the two probability measures merely agree on what events are possible/impossible. So, equivalent measures merely agree that any stock price in [0,infty), say, is possible and any negative price is impossible, and that's it.Gotcha. Thanks, Alan.Couldn't someone have come up with a different term that didn't already have a meaning?

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 20th, 2011, 3:53 pm

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 20th, 2011, 5:53 pm
QuoteOriginally posted by: numbersixPresumably, this fair value is the one allowing you to break even on average or in the long run. It is supposed to be objective and to exist out there in 'nature'. To my mind, this is a myth, the avatar of the myth of the 'fundamental value' of things. In any case, it depends on the existence of the long run, i.e. on the fact that the contingent event under scrutiny is part of a statistically series, either empirically or speculatively (i.e. as a thought experiment). Pricing by no-arbitrage is the same thing as pricing by the market. So the market is the one big difference between actuarial valuation and financial pricing. Because the market implies an equilibrium of supply and demand, it might well be the case that the market does not settle on pricing contingent claims at their actuarial 'fair value' (whatever it means). As a matter of fact, traders do not care about the long run and it is often the case that there is no such long run to begin with, because the majority of financial events, such as the default of a corporation, are one of a kind and are never part of a statistical series. So traders pay what they want to pay. Their only constraint is no arbitrage and this, by a famous theorem, implies that the market price of contingent claims should be expressed as the mathematical expectation of their payoff under some formal probability measure, discounted by the risk-free interest rate. This pricing operator is formally probabilistic in the sense that it is positive, additive, and sums up to one. It is not related to the real events of the real world in any historical or statistical sense. And why discount by the risk-free rate rather than some rate of return of the asset? Because the risk-free bond has to be priced with the same operator. "Fundamental" value or "fair" value don't have to be objective quantities (they certainly aren't objectively knowable anyway). However, anyone who trades on the assumption that they are completely unknowable will have no idea how to trade. I take it as self-evident that on aggregate at least traders operate as if they have such a value in mind, many individual traders (but perhaps not all) buying if they think an asset is under-priced and selling if they think it is over-priced. Isn't it implicit that such valuations lie behind "supply and demand" in purely financial markets (as opposed to trading goods and services).A similar argument applies to "objective" probability distributions. If one accepts that there is a range of possible future prices, and that some possibilities are more likely than others, then the idea of a probability distribution naturally arises -- but only as a way for us to describe the possible future. So I take it as trivially obvious that we will necessarily talk in terms of probabilities of future values and their implications for current values. The difficulty seems to me to lie purely in the ambiguity in identifying the conditional factors on which any distribution depends and that is where the ambiguity and confusion in pricing methods arises.

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 20th, 2011, 7:32 pm
You are saying that probabilities are in fact subjective probabilities. I don't think actuaries or BSM would agree.What I am saying is not that the objective value, or the objective probability, is unknowable; I am saying it doesn't exist. My problem is ontological, not epistemological.

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 20th, 2011, 10:56 pm
QuoteOriginally posted by: numbersixYou are saying that probabilities are in fact subjective probabilities. I don't think actuaries or BSM would agree.Perhaps, but that's their problem not mine.QuoteWhat I am saying is not that the objective value, or the objective probability, is unknowable; I am saying it doesn't exist. My problem is ontological, not epistemological.In the sense that the term "objective probability" is used to distinguish the "real world" case from risk-neutrality, I am saying it clearly does exist, if only as with every other human notion of reality, as a human concept that is employed every day, explicitly or implicitly, by traders in markets. However it does not necessarily exist unambiguously -- with objectively knowable qualities. So there are also two meanings of objective here that we need to be clear about.At the most trivial level, is it not at least as reasonable to claim the existence of objective probability as claiming that any form of objective physical measurement exists? We can't count anything without setting the conditions of what we are counting. If we count balls, how do we recognize a ball that we should count, since no two balls are ever exactly the same? The same goes with probabilities, we can always imagine the ability to count frequencies of identifiable future events as long as we also adequately identify the repeatable circumstances -- whether or not they ever repeat. In essence, what I am saying is that as long as we expect to be able to count, we can also expect counted frequencies to exist and that is sufficient to measure probability. So I find myself asking you: If you claim objective probability does not exist are you simply saying that nothing objective exists? How about price: Does an objective price exist? If so when? At the moment of a trade, perhaps? How about a millisecond later?It's trivially obvious to me that science can never make any objectively true statement about reality (as opposed to a logically consistent statement about our theory of reality) because every word that attempts to be descriptive of reality has an implicitly subjective meaning. But such subtleties tend to be irrelevant to life in most practical circumstances. We get on with science/finance where we usually intend "objective" to mean something more like "a situation uniquely consistent with an agreed overall view of reality". Is that not good enough for you? If it is, then it seems to me that the existence of an "objective" probability is merely a matter of agreeing the conditions under which we can imagine counting future events. And even when we don't agree such, it is still reasonable for an individual practitioner to imagine that a meaningful quantity could be agreed under the right circumstances and attempt to estimate it and it would also be reasonable for that practitioner to call it their estimate of the objective probability.

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 21st, 2011, 2:07 am
QuoteOriginally posted by: numbersixYou are saying that probabilities are in fact subjective probabilities. I don't think actuaries or BSM would agree.What I am saying is not that the objective value, or the objective probability, is unknowable; I am saying it doesn't exist. My problem is ontological, not epistemological.I am not sure I fully appreciate your point. Certainly we will never completely understand the market or completely know the underlying probability. I don't think that should stop me from trying to come closer to the underlying probably than you do and make some money.I am trying to understand the subtlety of your point from the viewpoint of a guy in the market paid to make models. If I am in the market and I rely on models, I don't necessarily think my model is "correct", just better than yours. For example, I don't think I know everything about Florida Gator football, but I bet I can come closer to their actual game results than you can (don't worry, college football and college basketball are probably the only two things which I would say that about), and part of that is that I have a better mental model than you do...definately not perfect, but better than most.

### Actuarial Pricing vs Financial Mathematics Pricing

Posted: June 21st, 2011, 7:18 am
Actuarial pricing only works if the insured events are independent. If I face a 5% probability of death in the next year, an insurer who can pool a large number of lives facing the same probability of death can provide me with insurance worth 20 times my annual premium and break even. If I am invested in the S&P500 and I face a 5% probability of losing 10% of my capital, actuarial methods cannot price the premium required to protect me from the loss. There is only one S&P500 index, if it crashes 10% every person in the risk pool loses at the same time and the premiums required to make them whole = 100% of the loss.