Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 1 Time Series.
 2 Classical statistics.
 3 Bayesian statistics.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Data Analysis.

ometime in the beginning of the year 2000 I was attending one of quantitative conferences in Houston. A lector was presenting a model for natural gas option prices that had good calibration properties but was transparently statistically unsound. I was an unexperienced quantitative analyst with first education in physics, hence, I could not resist to protest. The lector, one of the well known academicians in the field, did not argue merits of the case for very long. He turned his back on me and, as he was walking away, he raised his hand in a picturesque gesture and declared: "If you do not want to recalibrate your model then we do not have a common basis for discussion".

This raises an interesting question. Some traded instruments have great liquidity for large variety of contract parameters. Why not just use spline interpolation to price those? It would certainly be cost effective. The shortest sufficient (but incomplete) answer is the following: "We cannot produce a good quality explained PnL without a sound statistical model." Indeed, the explained PnL is the Taylor decomposition of the daily change in the portfolio's value. If the model is statistically unsound then the calibration parameters will swing and, without delta neutrality to those parameters, the explained PnL and hedging strategy would be useless. If the model is statistically sound then the calibration parameters would be stable and the noise parameters (those we delta hedge) would be as small as they can be. This way the daily Taylor decomposition (that we call "explained PnL") actually converges most of the time and the initial trade, executed at the derived from the model price, would not introduce a consistently bleeding position in the book.

It is important to note that nowhere in the above statements do we need the model to be of universe nature or to price the entire book uniformly with the same model. However, we do need orthogonal, clearly and uniformly defined sensitivities that we would delta hedge. Using a consistent universal model is sometimes harmful to purposes of trading and risk management. For any person with natural science background such idea seems absurd. To see that it makes perfect sense consider trading of vanilla options. The prices are given by the volatility smile and other observable parameters. There is simply no place for more information. However, if one trades forward start options then a model of forward volatility is required. Such model may reveal that vol-smile parameters do not follow a Markov process but the stat-arb strategy that would come from such discovery may still be prohibited by the cost of hedging against unlikely events. Another way to explain the same conclusion is to point out that financial derivatives formally depend on several market parameters but are motivationally constructed to be a bet on one market parameter. The rest of the parameters are hedged by other market instruments. Hence, modelling several derivatives with a single model is a futile attempt due to the competitive nature of the business. One model for each derivative and/or trading strategy always works better.

To summarize, we would like to separate all quantities of interest into two categories. First category is random by nature, we should be able to delta hedge against it. The parameters in the second category are not hedgeable but stable. Such task is hard enough. Hence, we again would like to introduce contract dependency into statistical modelling. We seek the simplest model that is suitable for the hedging purposes of the contract. The better model is the one with better stability and smaller noise terms. The pricing derives from the hedging.

Similar discussion may be found in the section ( Implementation tools II ).

 1 Time Series.
 2 Classical statistics.
 3 Bayesian statistics.
 Notation. Index. Contents.