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The Actuary The magazine of the Institute & Faculty of Actuaries

Arbitrage-free pricing

Many actuaries mistrust arbitrage-free pricing because they consider it inherently unrealistic. Frequently heard criticisms are:
– ‘arbitrage-free pricing does not allow an equity risk premium to be recognised’; and
– ‘arbitrage-free pricing does not allow mean reversion to be recognised’.
These assertions are often swiftly followed by affirmations of long-held beliefs:
– ‘so something sensible like the Wilkie model must be better’.
The first two statements are half-truths at best. They certainly do not justify the final remark. The third comment may still be a perfectly sound conclusion, but for reasons different from those proffered. In this article, we explore the reasons why.

Quantile reserves solve part (90%?) of the problem
Actuaries educated in traditional actuarial techniques are quite accustomed to calculating values of guarantees based on postulated real-world outcomes of various unknowns. However, traditional deterministic methods ignore the inherent spikiness of asset returns, missing the variability of both reinvestment and disinvestment risk. Adding arbitrarily large margins is at best crude, and there is little way of knowing if enough has been added.
To capture some of this asset variability, actuaries have for many years postulated probability density functions for some of these unknowns and then held reserves and capital to cover some quantile of the distribution of the values of the guarantee costs emerging. The Wilkie Model is probably the best-known and most widely used tool for this type of analysis.
These techniques allow for the actuaries’ best estimates of risk premiums over risk-free assets and for real-world probability distributions. The approach has the appeal of according with people’s intuition about asset returns and business cycles. In particular, it can be calibrated to deliver what are judged to be the right levels of paper millionaires (and paupers) over a given projection period. It can also allow for discrete (ie realistic) assumed rebalancing of investment portfolios as the markets move upwards or downwards.

Discretionary benefits
Calibrating such a model requires a considerable degree of judgement. While it appears that we have huge volumes of data from investment markets, if we wish to project over the term of our liabilities, we have scarcely any data. How many disjoint 25-year periods of data exist?
An equally thorny issue is that of selecting the level of quantile reserve, or probability of ruin. We suspect that an acceptable level of quantile reserve would need to be large to be acceptable to the FSA. But this pushes us to the very edge of our distribution, where we have the least certainty over the value of our results: the tail probabilities of distributions projected 25 years into the future.
Furthermore, no matter what quantile is chosen, there cannot be certainty that the provisions made for the guarantee are adequate. Arbitrage-free pricing is different: it aims to return a reserve that, if invested and dynamically managed according to some algorithm, will meet the value of a guarantee with certainty.

Bookmakers don’t go bust
To understand the operation of arbitrage pricing, an analogy is helpful. The fable of the bookmaker is widely used to illustrate the difference between arbitrage pricing and expected outcomes. Suppose a bookmaker is running a book on the outcome of tossing an unbiased coin, and he intends to make neither a profit nor a loss. The probability of a head is equal to the probability of a tail, so an actuary would say that the bookmaker should offer odds of evens on each outcome. Then, irrespective of how money is placed with the bookmaker, so long as the game is played enough times, the bookmaker can expect to come out even. But suppose gambler A bets £4 on heads and gambler B bets £2 on tails. If the bookmaker offers evens on each outcome, after one game, he will either be £2 up or £2 down. Over an infinite number of games, his expected profit will be:
(1/2*2) + (1/2*-2) = 0.
But it is quite possible that, before he gets to his expected loss, he will have gone bust. Indeed, since we know that a Brownian motion series is certain to cross any given level, if this game is played with the same stakes and at these odds for long enough, it is certain that, sooner or later, the bookmaker’s accumulated losses will wipe out his starting capital.
If, on the other hand, he had offered odds of 21 against tails and 21 on heads, he would be guaranteed to come out even after the game, irrespective of the outcome. And so long as he recalculates his odds to reflect the placement of bets, he can always guarantee to come out even.
So our bookmaker, never having heard of actuaries or the law of large numbers, ignores his knowledge of the real-world probabilities and sets his odds so as to hedge his bets. The odds he sets leave him uninterested in the actual outcome. We term the odds, or rather the probabilities implicit in the odds, ‘risk-neutral’.

Option prices for provisions
Arbitrage-free pricing, as practised in option markets, is based on similar principles. A simple example shows how option pricing techniques may be applied to the with-profits business.
Setting aside smoothing, the claim on a with-profits insurance policy with a maturity guarantee may be decomposed into two constituent parts:
– the asset share;
– a top-up consequent on the guarantee, ie max {0, guaranteed amount asset share}.
The payout on the second component has exactly the same definition as the claim amount on a put option with the:
– strike price equal to the guarantee;
– exercise date equal to the maturity date; and
– underlying asset equal to the asset share.
So we should be able to price with-profits guarantees in a similar manner to that in which the market prices options.

Myth: Risk premium and mean reversion are inconsistent with arbitrage-free pricing
Let us return to our bookmaker. Suppose we introduce an element of discounting by assuming the coin toss occurs one year after he has taken his bets and set his odds. If he is taking no risk, what rate should he use to discount his payouts? Our intuition would suggest a risk-free rate, and our intuition is right. If the bookmaker is consistent and wishes to be able to meet the payouts with certainty, he must invest in a risk-free asset and so he will earn a risk-free return. We note that this risk-free rate is consistent with the risk-neutral probabilities implicit in the odds the bookmaker offered: both are consequences of the bookmaker’s risk-aversion.
The analogy holds in the world of option pricing: risk-neutral probabilities are consistent with risk-free rates of return. That the method takes no account of risk premiums is a benefit to a regulator, for it removes one source of variation between companies. The actuary cannot, by assuming some expected risk premium, reduce the value of reserves simply by investing in riskier assets. Such an action would be tantamount to a ‘free lunch’.
Arbitrage-free pricing does not deny the existence of risk premiums (or mean reversion) in the real world. Both these features can be introduced into the drift parameters of the Brownian motions that assets are assumed to follow. But if we add back these properties, we have to change the probability distributions. If we did not, we would end up with the free lunch we are trying to eliminate for reserving purposes. The effect of adding back mean reversion into an arbitrage-free model is to end up at the same arbitrage-free price, albeit with rather more complicated mathematics.
The key point in the above is that risk-neutral methods are purely a computational device to get at the required arbitrage-free price. They are based on an artificial measure, not a real-world measure, so any statements of other model properties, eg absence of mean reversion, are statements about the model, not the real world. Not including such properties in a particular risk-neutral model does not invalidate the risk-neutral model.

Assured solvency
The beauty of these reserves is that there is no need to worry about not meeting liabilities. If you buy the underlying option or hedge your risks as the bookmaker would, the provision is certain to meet the guaranteed amounts, if the correct trading strategy is followed. But what is this correct trading strategy?
In simple cases, the Black-Scholes model tells us that for a put option, typically, the trading strategy will be to hold a short position in the asset and a long position in cash and for this portfolio to be continuously rebalanced. A life insurer’s case is much more complex, because the mix of the underlying asset is itself dynamic, as is the guarantee, but the similarity of the guarantee to a put option implies the trading strategy should look similar.

Actuarial redundancy
The method offers us the prospect of an objective measure of the value of a guarantee. There is only one arbitrage-free price for any security, and similarly, there will be only one for a with-profits guarantee. The element of judgement an actuary habitually exercises is to be rendered redundant.

Modern myth: Arbitrage-free prices are objective
Before we rush off and set up delta-hedging portfolios, making market makers very happy, let us acknowledge that there are problems with arbitrage-free methods. The methods require us to make an assumption about a whole host of parameters and correlations every bit as sweeping and significant as those used by die-hard quantile reservists. We will revisit this topic in the next article, satisfying ourselves now with an illustration using volatilities (see table 1 above).
Despite rumours to the contrary, quantile reserves and arbitrage-free prices are not mutually inconsistent: they are simply making different assumptions and allowing judgement to be exercised in different ways. Neither is perfect. Neither is wrong. In the next article we look at some of the practical issues associated with option pricing methodologies.