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Arbitrage-free pricing

As discussed in our previous article, arbitrage-free pricing offers the prospect of an objective measure of a with-profits fund’s guarantees, one that is not susceptible to manipulation by those secretive practitioners of the black arts: actuaries.
All we need to do is to set up an asset model to generate simulations from which the prices of traded options can be calculated. We must demonstrate that these calculated prices match those actually observed to within some multiple of the standard error. Then we can project our liabilities on the same simulations, prescribe future investment and bonus strategies, and calculate an expected value of the cost of guarantees.
The following questions arise:
1 Should our model be calibrated to the swap curve or to a gilt yield curve?
2 What option prices should we try to replicate from our simulations?
3 How can we deal with profit margins and other distortions?
4 How should we deal with short-term market movements?

Swap curve or gilt yield curve
Our valuation actuary needs a risk-free rate of interest in order to set up the appropriate risk-neutral measure and the required discount factors. There are two sets of market data that we can use to calibrate this curve: the gilt market and the swaps market. A simple investigation of the market shows that using the swaps market can result in considerably higher yields than gilts and commensurately lower option prices (see figure 1).
Before our actuary rushes off to use the swap spread we need to understand why it might exist. If the swap spread is a risk premium then it should be reduced accordingly.
Suppose our investor, wanting fixed-rate interest for five years, buys a swap of that term. He now needs to finance his floating rate interest payments, at say LIBOR. To do this, he must deposit his cash at floating rate with a bank for, say, 180 days. And after the first 180 days, and each subsequent period, he must roll his deposit over. He is therefore exposed to the risk of counterparty default in respect of that bank deposit. In a similar way, if interest rates were to fall markedly and the swap counterparty were to default then the swap would need to be set up again under considerably less advantageous terms. Fortunately, in most cases collateralisation will be used to mark the swap to the market each day, all but eliminating this source of counterparty risk.
The swap spread exists, in part at least, to compensate this investor for the risk from placing this investment on deposit. The swap curve cannot therefore be deemed risk-free. More precisely, the assessment of a swap as risk-free is equivalent to assessing LIBOR as risk-free. A deduction from the swap spread to allow for the default risk of the bank is needed before we can use it in our arbitrage-free price. But it is notoriously difficult to decide exactly what the deduction should be, even once we have identified the risk: it should at least cover the expected rate of default, but how should it allow for the uncertainty of future default rates?

Choice of options
We know the options should reflect the nature and term of our liabilities. It matters because some of the key parameters of the options, such as the implied volatility, vary considerably with term and the degree to which the option is in the money (or ‘moneyness’). Figure 2 illustrates the term structure. The often-observed variation of the volatility with the underlying strike price is referred to as the volatility smile. Our guarantees are from a wide spectrum of term outstanding and moneyness. There are two obvious problems:
– We may not be able to calibrate our model to every option that corresponds to a certain class or generation of liability.
– We may well have different classes of business at the same term outstanding, one where the option is deeply in the money, and one where it is not.
The issue of variable moneyness at a particular duration is particularly troublesome. Practical considerations mean that we do not want to run our simulations twice to capture option values for the in-the-money portfolio separately from the out-of-the-money portfolio. One apparently simple solution would be to pick a weighted-average implied volatility. This may prove tricky to determine as it requires a segmentation of liabilities by term outstanding and moneyness, and the calculation of trial values for the costs of guarantees attaching to each cell. A closed-form approach even to these trial calculations may not work well, because of the complexities introduced by future management actions and future premiums.
An alternative is to fit a more complicated asset model which allows instantaneous volatility to depend on both time and the underlying asset. With a sufficiently fine mesh of nodes (price and time), distinct implied volatilities can be supported from the same model at the same time. However, fitting such a model will not always be simple. A more pragmatic actuary may well resort to judgement.
Before our valuation actuary invests large amounts of time seeking to recreate a volatility smile, we ought to stop for a moment and ask whether we should try to replicate any particular smile in the first place. Does the smile reflect complicated volatility structures known only to skilled market makers, or is our actuary at risk of reserving making full allowance for those market-makers’ profit margins?

The volatility of volatility
We have already commented that the market exhibits a pronounced term structure of volatility. Figure 2 shows:
– historical implied volatility for the FTSE100 from Bloomberg; and
– the five-year term structure of implied volatilities from Merrill Lynch.
The figure reveals something new: the implied volatilities themselves exhibit considerable variation. Long-term options’ implied volatilities are usually lower, but occasionally higher, than shorter dated volatilities, and the implied volatility of a five-year option has moved widely, between approximately 15% and 30%. The implied volatilities of short-term options are themselves more volatile than those of longer-term options.
We might want to abandon the simple single-volatility rate Black-Scholes model in favour of something with a term structure. In practical terms, when we are generating our simulations all we need to know is the prevailing rate of volatility at time step t, say. We will soon hit the problem we encountered earlier when trying to use the one set of simulations to evaluate in-the-money and out-of-the money options simultaneously. Our simulation at time 12 months is being used to price an option vesting at time 12, but also one vesting at time 24 and so on. It’s getting complicated again.

Profit margins
Suppose our observed implied volatilities do not extend as far as our liabilities and do not cover the required range of moneyness. We reach an impasse. To be meaningful, we must calibrate to the observed implied volatility of traded options; but there is no liquid market in options of sufficiently long term and for options significantly in or out of the money. An illiquid market is subject to distortions and arbitrage: even call put parity can break down.
It is also worth recognising that market-makers try to extract a profit from selling options (see figure 3). Implied volatilities from option prices will have traders’ profit margins built in. While in the liquid market of short-term options we may assume that competitive pressures keep profit margins down at insignificant levels, in the less liquid markets of long-term options, the profit margins may be significant. In reality, a sizeable trade will see even wider profit margins emerge.
What is the sense of an arbitrage-free price that itself is calibrated to illiquid options that may not themselves be arbitrage-free? If we decide that the whole aim of market-consistent valuation is to price the transfer of risk, then we might conclude that we have no alternative but to calibrate to an illiquid and imperfect market. Alternatively, if the aim of the exercise is to hedge the risk ourselves, we should clearly set aside profit margins and instead make an allowance for the trading costs in this replicating portfolio. Estimating the profit margin is all but impossible. There appears to be no alternative to actuarial judgement again!

The bottom line
Putting aside the practical difficulty associated with calibrating a term-dependent moneyness dependent option model, we should pause for a moment to consider what our valuation actuary is letting himself in for. We have just identified that volatility is inherently volatile. Looking more closely, we can see that at moments of crisis, the differences between implied volatilities and historical volatilities can widen. It appears that by calibrating to implied volatilities, the solvency of a with-profits fund will be exposed to the market’s whims. Crucially, will life insurers start trading in ever larger amounts, adding to the volatility?
Alternatively, an insurer seeking to minimise exposure to an otherwise unpredictable and uncontrollable source of risk might conclude that the only rational solutions are to buy the portfolio of options, or to match guarantees almost completely. Is this really the intention of policymakers?

Looking forward to volatility
Does the term structure observed on the day of the valuation have any particular relevance to a long-term valuation? Should not the valuation be based rather on an average of observed implied volatilities over some lengthy period?
The latter reintroduces the prickly matter of actuarial judgement to determine what type of average, over what term, and so on. Most defensible would be the construction of forward rates set at market rates at the short liquid end and reverting to longer-term averages at the long end. This would offer a modicum of predictability and stability. The result of such a calculation that is not based on market-observable data is certainly not the reserve that will allow us to purchase options at the valuation date, nor is it certain any longer to be the reserve required to set up a replicating portfolio. But if we have already concluded there is little objectivity associated with such market consistency, the result might be acceptable.

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