Are long-dated options correctly priced?

One observation that strikes any beginner in the FX markets is that during trends, buying and holding options looks an easy winning deal. Does option pricing reflect this?

One of the main assumptions of the classic pricing models is that changes in financial assets are random variables.

This is the classic equation leading to options pricing models: dLn [ P(t) ] = [Mean]dt + [sigma]dz where dz is a brownian factor and t represents time.

One has to briefly examine in what conditions a trend could emerge given this formula. The first obvious answer is by having [Mean] equal to a non-0 factor that would be more or less stable during certain periods of time.

The only reason for stable structural trends in the G7 FX markets would come from long-term interest rates differentials, but this would not really support the magnitude of the moves.

Another way to witness trends would be if dz was not brownian -- if the 'noise' factor in the equation was auto-correlated. This contradicts one of the main assumptions behind market behavioural theory -- that markets are efficient -- meaning the entire available information is immediately reflected into asset price changes.

Most classical models are based on the idea that markets are efficient, which means the information available somewhere is available everywhere. This can be challenged, but asset behaviour does resemble the lognormal laws. Indeed, sensitive technical models are active in the market with potentially unlimited size of interventions. That means any unexplained disturbance in price fluctuations could be picked up, which would move the market to its efficient price level even if the actual information that triggered the first deals by those 'who knew' was still unknown to most of the players at the moment of the move.

Eventually, there is no reason why the unexpected components of series of news should be biased in the same direction during consistent periods of time, thus the 'noise' factor should not create trends. If markets were following such random behaviour, historical returns would fit with the expected lognormal patterns.

Over the past 30 years, FX implied option volatilities have roughly fluctuated between 7% and 15% as far as US$/DM and then euro/US$ are concerned, with an average over time around 10--11%. Normally, price changes should fall with a probability of 68.3% within a +/- standard deviation range around the predicted forward rate, with a 95.4% probability inside a +/- 2sd and 99.7% within a 3sd range.

Studying long-term time series, we find quite different numbers for the one-year changes in dollar rate against euro (see box).

With 46% of these percentages being above 10%, and 40% above 11%, we are well above expected numbers of 31.7%; even the 1.70% of occurrence of over 30% changes is a surprise when expecting just above 0.3%.

Option markets are aware of this propensity for unexpected large events to happen more than they should, and usually charge a higher volatility for smaller delta options, taking into account the 'fat tails' of the market distribution. When market-makers perceive a trend is in place, they assymetrically price this in the volatility structure, and this is called the skew.

The reason for market-makers to sell volatilities during these regular trends is that a delta-neutral short vega short gamma portfolio will make money during these periods of time since the lack of two-way movements will prevent significant losses arising from the delta management of the position.

Thus, from a market-maker point of view, being long options during regular ongoing trends is obviously a losing game, and volatilities have to go lower.

But for the classic final buyer, asset managers or corporates with real hedges, the conclusion is the opposite: one-year premium of 4% or so will look cheap compared with historical annual changes of 20%, 30% or more.

Thus the market-maker and the hedger could look at the same volatility as quite expensive or relatively cheaply priced.

This kind of situation occurred during 2003 as euro/US$ moved higher by more than 20% with average volatilities around 10%, and a regular trend taking place.

Thus the option market is able to find an equilibrium where different types of players will eventually meet, although some of them will see a bargain where others think prices are expensive.

This situation can only exist because we all agree on a model that is flawed from its very inception. Taking into account adaptive processes of asset prices, adjustment would lead us to use different models, where this is not the standard deviation of a supposedly lognormal distribution that we would be manipulating to adapt in the real world, but the average expectation of the relative changes of price during certain periods of time. This approach then supposes some level of inefficiency -- and provides the main theoretical justification for model hedge funds to exist: without inefficiency, there would be no loophole for them to prosper from!