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 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 1 Calculational Linear Algebra.
 2 Wavelet Analysis.
 3 Finite element method.
 4 Construction of approximation spaces.
 5 Time discretization.
 6 Variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## Implementation tools II.

here is a profitability threshold for every strategy of statistical arbitrage. Such threshold 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 money.

The above example is basic for understanding some general tendencies. If one uses linear stochastic models of limited dimensionality 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 sufficiently profitable 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 consistently profitable, acceptable from risk-management point of view (=sufficiently profitable to afford tight static hedges) and simple. Cheating-free way to profits passes through the domain of high-dimensional modelling. The more information is inputted into a model, the more profitable it might be. This is not a contradiction to discussion of the section ( Data Analysis ). For example, volatilities and correlations are known to have significant predictable component. Once such component is extracted via modelling and utilizing 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.

 1 Calculational Linear Algebra.
 2 Wavelet Analysis.
 3 Finite element method.
 4 Construction of approximation spaces.
 5 Time discretization.
 6 Variational inequalities.
 Notation. Index. Contents.