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ClosetChartist
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Joined: July 17th, 2003, 4:41 pm

### Actuarial Pricing vs Financial Mathematics Pricing

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.

numbersix
Posts: 474
Joined: July 23rd, 2001, 2:33 pm

### Actuarial Pricing vs Financial Mathematics Pricing

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?
Last edited by numbersix on June 15th, 2011, 10:00 pm, edited 1 time in total.

Marsden
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Joined: August 20th, 2001, 5:42 pm

### Actuarial Pricing vs Financial Mathematics Pricing

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.

numbersix
Posts: 474
Joined: July 23rd, 2001, 2:33 pm

### Actuarial Pricing vs Financial Mathematics Pricing

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.

Marsden
Posts: 3829
Joined: August 20th, 2001, 5:42 pm

### Actuarial Pricing vs Financial Mathematics Pricing

"Doesn't exist" isn't really correct: aren't there potentially thousands of "objective probabilities" for every contingent claim?

Alan
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### Actuarial Pricing vs Financial Mathematics Pricing

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.
Last edited by Alan on June 19th, 2011, 10:00 pm, edited 1 time in total.

Edgey
Posts: 223
Joined: March 23rd, 2005, 11:01 am

### Actuarial Pricing vs Financial Mathematics Pricing

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)

Verde
Posts: 9
Joined: December 18th, 2009, 11:49 am

### Actuarial Pricing vs Financial Mathematics Pricing

http://www.soa.org/library/newsletters/ ... 9903.pdfAn Actuary Looks at Financial Insurance By Richard Q. Wendt With the development of new variations of equity indexed annuities and complex pension benefit design, new forms of return guarantees have been created in both insurance and pension. These guarantees create a form of "financial insurance" for investors and plan participants, as the expected returns are protected against market downturns. Today, both insurance and pension actuaries are determining the cost of return guarantees and appropriate investment strategies to support those guarantees. Some actuaries may be surprised to find that the conditions necessary for the use of traditional actuarial analysis may not exist in financial insurance and that the costs developed by traditional actuarial analysis may significantly understate the "true" cost. This article discusses the differences between traditional actuarial analysis of financial insurance and an option based analysis. In 1995, Professor Zvi Bodie published his controversial article, "On the Risk of Stocks in the Long Run" in the Financial Analysts Journal. That article showed that the price of a put option on the S&P 500, with a strike price based on the risk-free rate of return, would increase with time horizon. That, Professor Bodie argued, indicates that stocks are indeed risky in the long run. Since his conclusion was contrary to the popular view of the long-term benefits of stock investment, his article generated quite a few responses. Bodie's article showed that the price of a 20-year put option, under stated assumptions, would be 34.5 percent of the initial investment. Several commentators (using the traditional analysis) stated that a premium of that level could not possibly be correct, since the probability of a twenty-year shortfall was given as only 4 percent. Option pricing concepts tend to puzzle and perplex actuaries and other analysts who are familiar with the more traditional insurance or probability-based approach. Thinking of a put option as a form of financial insurance, the traditional actuary would likely attack the problem of pricing a put option by determining the distribution of future results, applying an interest discount factor and then summing the probability-weighted present values of all outcomes. It often comes as a shock that the financial insurance premium is quite different than the option price calculated by the well-known Black-Scholes option formula. Many analysts are aware that the Black-Scholes option price is primarily determined by the volatility of the security, and know, but may not fully understand why, that the expected return on the security is not an element of the formula. On the other hand, the traditional insurance methodology primarily references the expected return, while the volatility of the results is not directly referenced. The two methodologies are quite different, so it should not be surprising that they may produce different results. First, some background on the Black-Scholes option pricing model. The B-S model makes the following key assumptions (in simplified form): ? The underlying security is continuously traded and can be bought or sold, in any amount. ? Cash can be borrowed or loaned at the risk-free rate, without limit. ? There are no taxes or transactions costs. ? Security prices are given by geometric Brownian motion. ? There is no arbitrage. Next, consider the key assumptions underlying an ideal insurance transaction: ? Large number of individual insured event ? Independence of insured events, in one year and over time. ? Predictable risk There are two key differences in the underlying assumptions, risk and a related issue, independence. Risk?the option transaction is priced to be arbitrage free, in the sense that an investor selling an option can, on the basis of the B-S model's stated assumptions, make continuous adjusting transactions in the underlying security to provide a guaranteed risk-free result. On the other hand, the financial insurance transaction is not completely risk free. To expand on the issue of risk in the financial insurance arena, recall that in Professor Bodie's model the risk of stocks performing below the risk-free rate is about 4 percent. One commentator suggested that, even if all value were assumed to be lost when there is underperformance, the cost of insurance would still be less than 4 percent (compared to the put option premium of 34.5 percent). Assume that an investor puts $100 in the stock market and also spends$4 for financial insurance. ? The financial insurance provides a benefit only if the 20-year stock return is less than the risk free rate. ? If the 20-year risk-free spot rate is assumed to be 6 percent, then the guaranteed amount would be $100 x 1.0620, which is$320.71. ? If the $4 insurance premium were to be invested at the risk-free rate, then it would accumulate to$12.83. How can $12.83 support a potential loss of$320.71? Only if, say, 100 investors buy the insurance and no more than four receive benefits. But if all 100 investors are investing in the same stock market at the same time, then either they all receive a large benefit or all receive no benefit. Given a stochastic future, the insurer would have a profit 96 percent of the time, but the other 4 percent of occurrences would be financial disasters. Whenever performance is poor and benefits are payable, the insurer would have only $13 of assets and a required payment of$321. Assuming no other resources for the insurer, there would be a 4 percent probability of bankruptcy. In fact, the risk-free insurance premium (i.e., the premium required to be 100 percent confident of covering all claims) would be 100 percent of the potential exposure (ignoring interest discount factors). The wonder of the B-S option pricing formula is that the true risk-free cost is much less than 100 percent. Some analysts might suggest that the insurance premium could be reduced by investing the premium in higher return equities. That would not improve the financial results, since the accumulated premium would itself be valueless in the 4 percent of scenarios where benefits were required. Independence?Suppose the insurer sells financial insurance policies every year, with each year's policies having a four percent expected loss. What premium would the insurer need to charge in order to assure solvency? Consider this simple example: ? 1000 policies sold each year for 25 years. ? The investment made under each policy is $1000 for a total of$1,000,000 in each year. ? The probability of loss for each policy after 20 years is four percent. ? The premium charged is four percent of the investment or a total of $40,000 per year. At the end of the first 20 years, the determination of potential claims for the very first purchasers is made. If there are claims (four percent probability), then$1,000,000 will be paid out, but total premiums only amount to $800,000 over 20 years of sales. Assuming no additional resources, the insurer can make only a partial payment to the initial purchasers, but is now insolvent and cannot provide any benefits to purchasers from years two through 20. It would seem clear that a four percent premium would be insufficient to provide reasonable assurances of solvency. But what level of premium would be required for a higher degree of security? Assuming that each issue year's results are independent for 25 years of issue, an eight percent premium would be required to cover the 92 percent probability of zero to two payout years in a 25-year period (assuming a binomial distribution). There would still be a remaining 8 percent chance of three or more payout years. For 98 percent certainty of covering payouts (up to three payout years), a 12 percent premium would be required. Notice that, even though thousands of investors are buying financial insurance, there are only 25 different insurable events, not 25,000. The traditional insurance requirement of large numbers of independent events does not hold. Furthermore, the payout years for this financial insurance would not actually be independent, since the investment periods overlap. For example, the admittedly extreme result of 100 percent loss in year 20 would wipe out investments for all purchasers in the first 20 years and trigger 20 years of payouts. In other words,$800,000 would be available to pay claims of \$20,000,000 (ignoring the risk-free accumulation of both premiums and claims). In order to evaluate the impact of overlapping periods, the Towers Perrin CAP:Link economic simulation model generated 500 stochastic scenarios covering 25 years of sales. That analysis indicated that a premium of 16 percent would be required for 90 percent certainty and a premium of 32 percent would be required for 98 percent certainty of covering all losses. When we adjust the financial insurance premium for the risk of insolvency, it is easy to see that an expected loss of four percent can be translated to a premium that approaches, or even exceeds, the Black-Scholes risk-free cost of 34.5 percent. This analysis shows that the traditional actuarial analysis of financial insurance may significantly understate the risk-free cost of the insurance and that more advanced techniques are needed to properly price the cost of guaranteed returns for equity-linked products or pension plans. Richard Q. Wendt, FSA, CFA is Principal at Towers Perrin in Philadelphia, PA. ________________________________________ References 1. Bodie, Zvi. 1995. "On the Risk of Stocks in the Long Run." Financial Analysts Journal, vol. 51, no. 3 (May/June):18-22. 2. Dempsey, Mike, Robert Hudson, Kevin Littler and Kevin Keasey. 1996. "On the Risk of Stocks in the Long Run: A Resolution to the Debate?" Financial Analysts 3. Journal, vol. 52, no. 5 (September/October):57-62. 4. Campbell, John Y., Andrew W. Lo and A. Craig MacKinlay. 1997. The Econometrics of Financial Markets. Princeton, NJ: Princeton University Press. 5. Ferguson, Robert and Dean Leistikow. 1996. "On the Risk of Stocks in the Long Run: A Comment." Financial Analysts Journal, vol. 52, no. 2 (March/April):67-68. 6. Financial Analysts Journal. 1996. "Letters to the Editor" from George M. Cohen, P. de Fontenay, Gordon L. Gould, and (response) Zvi Bodie, vol. 52, no. 2 (March/April):72-76. 7. Mulvey, John M. 1996. "Generating Scenarios for the Towers Perrin Investment System." Interfaces, (March/April):1ff. 8. Mitchell, G. Thomas and John Slater, Jr. 1996. "Equity-Indexed Annuities?New Territory on the Efficient Frontier." Product Development News. Issue 39 (January):1ff. 9. Ruark, T. J. 1996. "Variable Annuities: What Goes Up, Must Come Down." Contingencies, (March/April):35-37. 10. Siegel, Jeremy J. 1994. Stocks for the Long Run: A Guide to Selecting Markets for Long-Term Growth; Burr Ridge, IL: Irwin 11. Taylor, Richard and Donald J. Brown. 1996. "On the Risk of Stocks in the Long Run: A Note." Financial Analysts Journal, vol. 52, no. 2 (March/April):69-71.

Marsden
Posts: 3829
Joined: August 20th, 2001, 5:42 pm

### Actuarial Pricing vs Financial Mathematics Pricing

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?

numbersix
Posts: 474
Joined: July 23rd, 2001, 2:33 pm

### Actuarial Pricing vs Financial Mathematics Pricing

Last edited by numbersix on June 19th, 2011, 10:00 pm, edited 1 time in total.

Fermion
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Joined: November 14th, 2002, 8:50 pm

### Actuarial Pricing vs Financial Mathematics Pricing

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.

numbersix
Posts: 474
Joined: July 23rd, 2001, 2:33 pm

### Actuarial Pricing vs Financial Mathematics Pricing

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.

Fermion
Posts: 4486
Joined: November 14th, 2002, 8:50 pm

### Actuarial Pricing vs Financial Mathematics Pricing

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.
Last edited by Fermion on June 20th, 2011, 10:00 pm, edited 1 time in total.

Beachcomber
Posts: 140
Joined: May 25th, 2004, 5:56 pm

### Actuarial Pricing vs Financial Mathematics Pricing

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.

Verde
Posts: 9
Joined: December 18th, 2009, 11:49 am

### Actuarial Pricing vs Financial Mathematics Pricing

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.