here is a profitability treshhold
for every strategy of statistical arbitrage. Such treshhold is dictated by
prices of instruments that may be used as static hedges against the residual
risks associated with the strategy.
Consider the following example. Historical data in every market suggests that
implied volatility is consistently higher then historical volatility.
Therefore, one may attempt to sell at-the-money vanilla options and
delta-hedge. Theoretically, such strategy should be profitable. In reality,
periodic extreme moves of the market prices are likely to result in losses. To
avoid such losses one may attempt to modify the strategy to sell at-the-money
options and cover the exposure to extreme price moves by buying offsetting
out-of-money options (this is what we call static hedges) and then delta
hedge. In practice, the volatility smile (and bid-ask spreads around it)
prevents profit. In fact, this argument may be used to calibrate the
volatility smile. Faced with insufficient profit, one may choose to ignore the
risk of extreme price moves and proceed without static hedges. Such strategy
is practically and conceptually inferior to selling the static hedges and then
doing nothing while hoping that the sold contracts would never come into
The above example is basic for understanding some general tendencies. If one
uses linear stochastic models of limited dimesionality to construct strategies
of statistical arbitrage then, regardless of large variety of details and
motivations, the resulting statistical arbitrage strategies are similar.
Initially, when only few people are doing it, such strategies are profitable
enough to buy static hedges. Eventually, as more people join the same
activity, the profit rate goes down but expectations of investors and level of
competition rise, people relax static hedges or stop installing them
completely and increase leverage. This leads to repeatedly observed situations
when small trouble in one asset class seems to bring entire market down. For
example, consider a situation of several well diversified and moneyed hedge
funds doing similar strategies with high degree of leverage. Some portions of
the position may be illiquid or too large to sell. If a single portion of such
portfolio suffers losses then, in leveraged situation and in absence of static
hedges, one has no choice but to raise cash by liquidating profitable and
liquid positions. If several major players are doing the same then large scale
concurrent selling occurs and the self-feeding cascading effect creates
repeatedly observed situation of global distress.
Fortunately or otherwise, there are no miracles. One cannot invent a strategy
that would be both consistenly profitable, acceptable from risk-management
point of view (=profitable enough to afford tight static hedges) and simple.
The cheating-free way to profits passes through the domain of high-dimensional
modelling. The more information is inputed into a model, the more profitable
it might be. This is not a contradiction to discussion of the section
). For example, volatilities
and correlations are known to have significant predictable component. Once
such component is extracted via modelling and utilising large range of other
market parameters, the remaining noise component is significantly smaller.
This leads to more stable dynamic hedges and, thus, lower derivative prices.
It also leads to multidimensional parabolic PDE problems. The combination of
technologies directed at solving such problems in real time is the key
implementation tool and the subject of the following chapters.