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Edgey
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Actuarial Pricing vs Financial Mathematics Pricing

June 21st, 2011, 8:10 am

QuoteActuarial pricing only works if the insured events are independent. Independence makes the calculations easier, but dependence can be included into the actuarial approach. QuoteThere 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.You could argue that the S&P is independent over time. The premiums from the weeks where there was no crash cover the time when it does. Or a more actuarial example. An insurance company specializes in insuring in the Gulf of Mexico. Most years it just collects the premiums, but every so often a bad hurricane appears and they get a lot of correlated claims.
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Marsden
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June 21st, 2011, 2:57 pm

QuoteOriginally posted by: numbersixBlack-Scholes-MertonIn case the objective underlying stochastic process you have assumed is Brownian motion, it so happens that the replication is perfect, i.e. the standard deviation of the P&L of the portfolio formed by the contingent claim and its dynamic hedge is exactly zero. This implies the price of contingent claim is uniquely determined. On the other hand, it also happens, by Girsanov theorem, that the choice of risk-neutral probability measure is restricted to one, because the change of the objective probability into an equivalent one -- 'equivalent' in the sense that Alan has just clarified, of course -- preserves the volatility of the Brownian motion and because the real drift of the underlying is in any case changed into the risk-free rate.To sum up, if you wish to price contingent claims in a market, under the hypothesis of Brownian motion as real underlying dynamics, the price has to come up equal to BSM independently of dynamic hedging. It is only that the capacity to trade the underlying and the contingent claim freely (what pricing in the market means) combined with the necessity to assign a zero price to impossible events (what the equivalent measure means) coincides with dynamic hedging.I don't follow this. Black-Scholes made their argument based on dynamic hedging, and offered an alternative (but not well-developed) argument based on CAPM. You claim, however, that the BSM price comes up independent of dynamic hedging. Is this from your reference to the Girsanov Theorem?
 
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numbersix
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June 21st, 2011, 3:15 pm

Yes, risk-neutral pricing assumes the risk-neutral measure is equivalent to the real one (Brownian motion in this case). Girsanov's theorem provides, if I am not mistaken, that the underlying dynamics is also Brownian motion under the change of measure, with the same Brownian volatility as the original one. On the other hand, you know the original drift has to be changed into the risk-free rate by definition or risk-neutral pricing. Therefore, option prices under Brownian motion are the mathematical expectation of the payoff under Brownian motion of volatility sigma and drift r, discounted by the same r.This is another proof of BSM pricing, if you will. I know BSM based their argument on perfect dynamic hedging under Brownian motion; this what I argue in my post. What I am claiming is that risk-neutral pricing is more general than pricing by dynamic hedging (as it only assumes no-arbitrage: think of incomplete markets where you might not hedge perfectly or even hedge at all), and that they merely coincide under Brownian motion, because the family of risk-neutral probability measures is reduced to one.
 
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Marsden
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June 21st, 2011, 7:53 pm

Are you able to get, though, the singular risk-neutral probability measure of BSM -- where the volatility of the "real" underlying stochastic process is preserved -- without the dynamic hedging argument? That's what I'm not seeing.
 
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numbersix
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June 21st, 2011, 8:14 pm

I must confess I am intrigued by this conclusion too. I used to think -- and even wrote, in one of my Wilmott articles -- that if dynamic hedging were not permitted, then the BSM formula with any value whatsoever of volatility, in particular a value different from the volatility of the 'real' process, would nevertheless produce derivative prices that are arbitrage free, because it is a priicng operator with all the required properties. I used to think that only dynamic hedging would impose that the volatility number in the BSM formula should be equal to the one of the real process. Until somebody mentioned Girsanov's theorem. So it must be that the BSM formula with a volatilty number different from the real one does not qualify as pricing under a measure that is equivalent to the real one. This would imply pathologies such as assigning positive price to impossible events. This is why I have to conclude that the argument of dynamic hedging under Brownian motion is logically equivalent to the conjuncton of the following two requirements: risk-neutral pricing (i.e. no arbitrage pricing) + preseriving equivalence of measures.
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Marsden
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June 21st, 2011, 9:14 pm

I'm trying to track down the implications of Girsanov's Theorem. What I have found most interesting is the following, from this very forum:QuoteOriginally posted by: HATherefore, if one is to construct a Wiener measure that is absolutely continuous with respect to another Wiener measure, the only choice is perturbing the path by a process with zero quadratic variation and with finite expectation, which is more or less adjusting the drift. You can't change the time scale, ... you can't alter the dispersion.But I'm thinking the interest in being "absolutely continuous with respect to another Wiener measure" probably comes from our old friend, dynamic hedging. So, using risk-neutral volatility Y where the underlying volatility is X would, for example, indicate delta hedge ratios that don't really create delta hedges.numbersix, you indicate a stronger (in my opinion) contradiction: "assigning positive price to impossible events." Are you certain of that, and can you give an example?
 
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Alan
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June 22nd, 2011, 3:14 am

If I can butt in, again, a possible event in a idealized BSM world, is that the instantaneous variance of return (at time t) = a,where 'a' is a fixed, given positive number.So, suppose a = sig1^2 in the "real-world" makes the event "possible/true", but a = sig2^2 in the "risk-neutral world" makes the event "possible/true", where sig1 <> sig2. (<> means 'not equal').So, the risk-neutral measure would have to say that a = sig1^2 was impossible, since by assumption the volatility is constant. In other words, the two measures would not be equivalent, as they would disagree on whether or not theevent E: Inst variance(t) = sig1^2 was possible or not.This argument relies on the fact that the instantaneous variance at time t can be measured to arbitrary precision by recording thestock price path {S(t)} in the vicinity of t. In contrast, this type of measurement will not pin down the drift coefficient, even if you record {S(t)} exactly over [0,T] where T < infinity.(see, for example, Merton's comment following (A.3) on pg 356, here) Anyway, this is one example that comes to mind for numbersix's comment about "assigning positive price to impossible events." It's just heuristic because we are matching a real number, admittedly an operational stretch.
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Verde
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June 22nd, 2011, 12:42 pm

QuoteOriginally posted by: EdgeyQuoteActuarial pricing only works if the insured events are independent. Independence makes the calculations easier, but dependence can be included into the actuarial approach. QuoteThere 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.You could argue that the S&P is independent over time. The premiums from the weeks where there was no crash cover the time when it does. Or a more actuarial example. An insurance company specializes in insuring in the Gulf of Mexico. Most years it just collects the premiums, but every so often a bad hurricane appears and they get a lot of correlated claims.The concept of insurance relies on the law of large numbers, which in turn requires a large number of independent events. To put is as simply as I can ? a life insurer with a risk pool of 1 person requires an actuarial reserve = the insured amount. An insurer with a pool comprising an infinite number of lives needs to annually accumulate a reserve = probability of death current year*insured amount.To survive insurers have to be able to offer competitive premiums and convince their customers that their actuarial reserves will be sufficient to cover the worst case claims scenario.I will use your example, but make simplifying assumptions to illustrate the point. Ignore interest.Hurricane insurer A has an infinitely large pool of clients situated in an infinite number of hurricane prone areas. It is known with certainty that hurricane claims will amount to 10% of the insured value every year. The law of large numbers allows the insurer to set premiums at 10% of insured value per annum to settle all claims in full. No further actuarial reserve is required. Insurer B only writes policies on properties in the Gulf. It knows for certain that out of any 10 year period 9 years will see no claims, but 1 year will experience claims of 100% the insured value. If it charges the same premium as insurer A, it will have no problem if the event occurs in year 10 when the accumulated reserves can cover claims. But they will not have enough reserves if the event occurs in any other year.They either have to: 1. Charge the entire 10 year premium (= insured amount) upfront -they will get no business guaranteed.2. Maintain a reserve equal to the insured amount (at a cost) which makes them uncompetitive. 3. Default on at least part of their obligations 90% of the time. Not a sustainable business plan.It is possible for a Gulf of Mexico specialist to survive, but they will have to use re-insurance, to effectively diversify their risk pool as much as the competitor.Actuarially determined market crash insurance premiums face exactly the same issue. There is some independence which can be exploited using different world markets, but the law of large numbers does not come into play. After maximum risk pooling you end up with the world index. The actuarial reserve required to settle the worst case claim scenario = the insured amount. You don?t need an insurance company to provide this product, just keep your premium in the bank.
 
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Marsden
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June 22nd, 2011, 1:07 pm

Alan, I think what you're saying is basically this: if our model says that a process follows (r,σ ) when it really follows (μ,σ ) our instantaneous "errors" will be of magnitude dt; but if our model says that a process follows (μ,s ) when it really follows (μ,σ ) then our instantaneous "errors" will be of magnitude sqrt(dt) -- much larger.This is fairly directly significant for dynamic hedging, but if that's the only reason, then there's sort of a circular argument: BSM is proved through an argument based on dynamic hedging; and dynamic hedging is possible only with a risk-neutral probability measure that follows BSM in preserving the underlying volatility. It's an attractive consistency, but it doesn't yet rule out other models, as far as I can tell.
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Edgey
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June 22nd, 2011, 1:11 pm

@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.
 
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Alan
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June 22nd, 2011, 2:28 pm

QuoteOriginally posted by: MarsdenAlan, I think what you're saying is basically this: if our model says that a process follows (r,σ ) when it really follows (μ,σ ) our instantaneous "errors" will be of magnitude dt; but if our model says that a process follows (μ,s ) when it really follows (μ,σ ) then our instantaneous "errors" will be of magnitude sqrt(dt) -- much larger.This is fairly directly significant for dynamic hedging, but if that's the only reason, then there's sort of a circular argument: BSM is proved through an argument based on dynamic hedging; and dynamic hedging is possible only with a risk-neutral probability measure that follows BSM in preserving the underlying volatility. It's an attractive consistency, but it doesn't yet rule out other models, as far as I can tell.Numbersix's point, if I understood it, was that, starting with some GBM, no-arbitrage forces you to consider equivalentmeasures. But, the only equivalent measures describe other GBM's with the same volatility. I was trying to give a heuristicexplanation for this well-known "identity of volatilities". Namely P ~ Q (where '~' means equivalence), forces the measures toagree on which events are possible/impossible. Now, focus on 'volatility' events: i.e. the event that the measured volatility = a.The measures have to agree on the possibility/impossibility of these.
 
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Marsden
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June 22nd, 2011, 2:39 pm

Then I guess I don't follow. It seems to me that a different volatility will neither eliminate the possibility of any outcomes under an original process nor create any new possible outcomes that did not exist under the original. Doesn't that fit the definition of "equivalence" as you have been using it? What am I missing?
 
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Alan
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June 22nd, 2011, 2:59 pm

Think of an event as anything we can make a bet about (in this idealized continuous-time world).We can make bets about the realized volatility, either instantaneous or cumulative. So, if youbelieve the volatility is different under the pricing process ("Q"), you will make bets reflecting that -- but always lose.Eventually, you will conclude the fair price of these different vol bets are 0 -- i.e. the outcome is impossible. I earlier said equivalence forced agreement that stock prices were non-negative. But it doesn't mean _only_that. It means agreement on which events, in the sense of all events, are possible or impossible. Basically, an event in (0,T) is anything at all we could bet about which would be settled byobserving the continuous stock price path during (0,T) and perhaps doing some calculations with that. This includes volatility calculations.
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Marsden
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June 22nd, 2011, 3:08 pm

All right, I can see that. But it assumes continuity.
 
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Alan
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June 22nd, 2011, 3:11 pm

Sure, but this idealized world is continuous.
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