Numerical Software, Market Data and Extreme Events Robert Tong

1. Numerical Software, Market Dataand Extreme EventsRobert Tong

2. Outline • Market data • Pre-processing • Software components • Extreme events • Example: wavelet analysis of FX spot prices • Implications for software design

3. Market data • Tick – as transactions occur, high frequency, irregular in time quote/price with time stamp • Sample tick data at regular times – minute, hour, day, … – low-high price • Bid-ask pairs – FX spot market • Time series – construct from sampled and processed data

4. FX spot market prices - USD-CHF • ticks (e.g. see www.dailyfx.com) • minutes • hours From: www.dailyfx.com/charts

5. Data cleaning Required to remove errors in data – • inputting errors • test ticks to check system response • repeated ticks • copying and re-sending of ticks • scaling errors How can false values be reliably identified and rejected ? • what assumptions must be imposed? • elimination of outliers based on an assumed probability distribution

6. Pre-processing • Tick data irregular in time – construct homogeneous time series by interpolation: linear, repeated value • Bid-ask spread – use relative spread • Remove seasonality • Account for holidays • Must not introduce spurious structures to data

7. Software components

8. Implementation issues Algorithm design – • Stability • Accuracy • Exception handling • Portability • Error indicators • Documentation These are independent of the problem being solved

9. Extreme events • Weather – storm • Warfare – explosion • Markets – crash Software – How should it respond to the unpredictable? What is the role of software when its modelling assumptions break down?

10. An illustration – another type of bubble Underwater explosions are used to destroy ships – the initial shock is expected and often not as damaging as the later gas bubble collapse. Left: raw data from sensitive, but un-calibrated pressure gauge Right: calibrated gauge uses averaging to produce smooth curve Use of averaging obscures critical event in this case.

11. Example: wavelet analysis of FX spot prices • Wavelet transforms provide localisation in time and frequency for analysis of financial time series. • This is achieved by scaling and translation of wavelet basis. • Decompose time series, by convolution with dilated and translated mother wavelet, or filter, • Discrete (DWT) Orthogonal Filter pair: H – high pass, G – low pass followed by down-sampling

12. Wavelet filters Family of filters by scaling Daubechies D(4) wavelet filters result from sampling a continuous function

13. Multi-Resolution Analysis Discrete Wavelet Transform (DWT) d1 d2

14. DWT implementation Orthogonal wavelet transform uses • filters defined by sequences: , • satisfying: , , • This allows for a number of variants in implementation numerical output from different software providers is not identical

15. Discrete Wavelet Transform – Multi-Resolution Analysis For input data , length , produces representation in terms of ‘detail’ and ‘smooth’ wavelet coefficients of length Uses • Data compression – discard coefficients • De-noising Disadvantages • Difficult to relate coefficients to position in original input • Not translation invariant – shifting starting position produces different coefficients

16. Maximal Overlap Wavelet Transform (MODWT)(Stationary Wavelet Transform) • Convolution:wavelet filters as in DWT • No down-sampling • MRA produces N coefficients at each level • Requires more storage and computation • Not orthonormal Advantages • Translation invariant • Can relate to time scale of original data • Does not require length(x) =

17. Choice of wavelet filter • Short can introduce ‘blocking’ or other features which obscure analysis of data • Long increases number of coefficients affected by ends of data set • Basis Pursuit seeks to optimise choice of wavelet at each level but requires more computation

18. FX: USD, GBP, EUR – NZD12 noon buying rates, Jan – Jul 2007

19. FX: JPY, USD, GBP, EUR – NZD12 noon buying rates, Jan – Jul 2007(from www.x-rates.com)

20. JPY-NZD, LA(8), MODWT(includes boundary effects) x(t) d1 d2 d3 d4

21. JPY-NZD, LA(8) MODWT(includes boundary effects) x(t) d5 d6 s6

22. Boundary conditions – end extension • Wavelet transform applies circular convolution to data • What happens at the ends of the data set? • End extension techniques – periodic reflection – whole/half-point pad with zeros • Boundary effects contaminate wavelet coefficients software should indicate where output is influenced by end extension

23. End extension Periodic Whole-point reflection

24. USD-NZD, Haar, MODWT Periodic end extension Level 1 detail coefficients Level 2 detail coefficients

25. USD-NZD, Haar, MODWT(end effects removed) x(t) d1 d2 d3

26. USD-NZD, Haar MODWT(end effects removed) x(t) d4 d5 d6 s6

27. Wavelet analysis for prediction • Extrapolationfrom present to near future is useful • Apply wavelet filters to for avoiding boundary effect • Select wavelet scales to identify trend and stochastic parts of data set • Use wavelet coefficients to compute prediction (see Renaud et al., 2002)

28. Implications for software development • Reproducibility is desirable – algorithms precisely defined to allow independent implementations to produce identical results • Edge effects – contaminate ends of transform for finite signals – software must indicate coefficients affected • Smoothing/averaging – software should indicate when underlying assumptions likely to be invalid • Pre-processing – ensure that structure is not introduced by interpolation to give homogeneous data set

29. Implications for software development • For extreme events – must not obscure or remove data relevant to critical events by averaging, smoothing, filtering.